Câu hỏi của Trần Minh Quân

Question

giải phương trình sau: $\sqrt{2x^2+3x+2}-\sqrt{x^2+x+1}=\sqrt{x}$

Phong · Accepted Answer

`\sqrt{2x^2+3x+2}-\sqrt{x^2+x+1}=\sqrt{x}(x>=0)``<=>\sqrt{2(x^2+x+1)+x}-\sqrt{x^2+x+1}=\sqrt{x}`Đặt: `\sqrt{x^2+x+1}=a;\sqrt{x}=b(a,b>=0)`Khi đó ta được:`\sqrt{2a^2+b^2}-a=b``<=>2a^2+b^2=(a+b)^2``<=>2a^2+b^2=a^2+2ab+b^2``<=>2a^2=a^2+2ab``<=>a^2-2ab=0``<=>a(a-2b)=0``TH1:a=0(tm)``=>\sqrt{x^2+x+1}=0``=>x^2+x+1=0` (vô lý)`TH2:a-2b=0``<=>a=2b``<=>\sqrt{x^2+x+1}=2\sqrt{x}``<=>x^2+x+1=4x``<=>x^2-3x+1=0``\Delta=(-3)^2-4*1*1=5>0``x_1=(3+\sqrt{5})/2`(tm)`x_2=(3-\sqrt{5})/2`(tm)Vậy: `...`Câu trả lời: `\sqrt{2x^2+3x+2}-\sqrt{x^2+x+1}=\sqrt{x}(x>=0)``<=>\sqrt{2(x^2+x+1)+x}-\sqrt{x^2+x+1}=\sqrt{x}`Đặt: `\sqrt{x^2+x+1}=a;\sqrt{x}=b(a,b>=0)`Khi đó ta được:`\sqrt{2a^2+b^2}-a=b``<=>2a^2+b^2=(a+b)^2``<=>2a^2+b^2=a^2+2ab+b^2``<=>2a^2=a^2+2ab``<=>a^2-2ab=0``<=>a(a-2b)=0``TH1:a=0(tm)``=>\sqrt{x^2+x+1}=0``=>x^2+x+1=0` (vô lý)`TH2:a-2b=0``<=>a=2b``<=>\sqrt{x^2+x+1}=2\sqrt{x}``<=>x^2+x+1=4x``<=>x^2-3x+1=0``\Delta=(-3)^2-4*1*1=5>0``x_1=(3+\sqrt{5})/2`(tm)`x_2=(3-\sqrt{5})/2`(tm)Vậy: `...`

Hoàng Minh Khoa · Answer

Step   2x2+3x+2−x2+x+1=xthe square root of 2 x squared plus 3 x plus 2 end-root minus the square root of x squared plus x plus 1 end-root equals the square root of x end-root2𝑥2+3𝑥+2√−𝑥2+𝑥+1√=𝑥√        2x2+3x+2−x2+x+1=xmodified the square root of 2 x squared plus 3 x plus 2 end-root minus the square root of x squared plus x plus 1 end-root equals the square root of x end-root with boxed outline2𝑥2+3𝑥+2√−𝑥2+𝑥+1√=𝑥√   Square both sides of the equation    2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+x+1=xmodified 2 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus x plus 1 equals x with boxed outline2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+𝑥+1=𝑥       Step   2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+x+1=x2 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus x plus 1 equals x2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+𝑥+1=𝑥       2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+x+1=x2 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared modified positive x with boxed outline plus 1 equals modified x with boxed outline2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+𝑥+1=𝑥   Cancel equal terms on both sides of the equation   2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus 1 equals modified 0 with boxed outline2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+1=0     Step   2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+1=0       2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of modified open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren with boxed outline end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+1=0   Simplify the expression   2x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of modified 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 with boxed outline end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2+𝑥2+1=0     Step   2x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+𝑥2+1=0       2x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+x2+1=0modified 2 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus x squared with boxed outline plus 1 equals 02𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+𝑥2+1=0   Collect like terms   3x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+1=0modified 3 x squared with boxed outline plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus 1 equals 03𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+1=0     Step   3x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+1=03 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus 1 equals 03𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+1=0       3x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+1=0modified 3 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus 1 with boxed outline equals 03𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+1=0   Add the numbers   3x2+3x+3−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus modified 3 with boxed outline minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√=0     Step   3x2+3x+3−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√=0       3x2+3x+3−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of modified 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed with boxed outline plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2=0   Collect like terms   3x2+3x+3−22x4+5x3+2x2+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus modified 5 x cubed with boxed outline plus 2 x squared plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+2𝑥2+3𝑥2+3𝑥+2𝑥2+2𝑥+2=0     Step   3x2+3x+3−22x4+5x3+2x2+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 2 x squared plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+2𝑥2+3𝑥2+3𝑥+2𝑥2+2𝑥+2√=0       3x2+3x+3−22x4+5x3+2x2+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of modified 2 x to the fourth power plus 5 x cubed plus 2 x squared plus 3 x squared plus 3 x plus 2 x squared with boxed outline plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+2𝑥2+3𝑥2+3𝑥+2𝑥2+2𝑥+2=0   Collect like terms   3x2+3x+3−22x4+5x3+7x2+3x+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus modified 7 x squared with boxed outline plus 3 x plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+3𝑥+2𝑥+2=0     Step   3x2+3x+3−22x4+5x3+7x2+3x+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 3 x plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+3𝑥+2𝑥+2√=0       3x2+3x+3−22x4+5x3+7x2+3x+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of modified 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 3 x plus 2 x with boxed outline plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+3𝑥+2𝑥+2=0   Collect like terms   3x2+3x+3−22x4+5x3+7x2+5x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus modified 5 x with boxed outline plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2=0     Step   3x2+3x+3−22x4+5x3+7x2+5x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=0       3x2+3x+3−22x4+5x3+7x2+5x+2=0modified 3 x squared plus 3 x plus 3 with boxed outline minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals modified 0 with boxed outline3𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=0   Move the expression to the right-hand side and change its sign   -22x4+5x3+7x2+5x+2=-3x2−3x−3negative 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals modified negative 3 x squared minus 3 x minus 3 with boxed outline−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=−3𝑥2−3𝑥−3     Step   -22x4+5x3+7x2+5x+2=-3x2−3x−3negative 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals negative 3 x squared minus 3 x minus 3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=−3𝑥2−3𝑥−3       -22x4+5x3+7x2+5x+2=-3x2−3x−3modified negative 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals negative 3 x squared minus 3 x minus 3 with boxed outline−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=−3𝑥2−3𝑥−3   Square both sides of the equation   4(2x4+5x3+7x2+5x+2)=9x4+9x2+9+18x3+18x2+18xmodified 4 open paren 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 close paren equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x with boxed outline42𝑥4+5𝑥3+7𝑥2+5𝑥+2=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥     Step   4(2x4+5x3+7x2+5x+2)=9x4+9x2+9+18x3+18x2+18x4 open paren 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 close paren equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x42𝑥4+5𝑥3+7𝑥2+5𝑥+2=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥       4(2x4+5x3+7x2+5x+2)=9x4+9x2+9+18x3+18x2+18xmodified 4 open paren 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 close paren with boxed outline equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x42𝑥4+5𝑥3+7𝑥2+5𝑥+2=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥   Distribute    444through the parentheses   8x4+20x3+28x2+20x+8=9x4+9x2+9+18x3+18x2+18xmodified 8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 with boxed outline equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥     Step   8x4+20x3+28x2+20x+8=9x4+9x2+9+18x3+18x2+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥       8x4+20x3+28x2+20x+8=9x4+9x2+9+18x3+18x2+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals modified 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared with boxed outline plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥   Collect like terms   8x4+20x3+28x2+20x+8=9x4+27x2+9+18x3+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals 9 x to the fourth power plus modified 27 x squared with boxed outline plus 9 plus 18 x cubed plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+27𝑥2+9+18𝑥3+18𝑥     Step   8x4+20x3+28x2+20x+8=9x4+27x2+9+18x3+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals 9 x to the fourth power plus 27 x squared plus 9 plus 18 x cubed plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+27𝑥2+9+18𝑥3+18𝑥       8x4+20x3+28x2+20x+8=9x4+27x2+9+18x3+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals modified 9 x to the fourth power plus 27 x squared plus 9 plus 18 x cubed plus 18 x with boxed outline8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+27𝑥2+9+18𝑥3+18𝑥   Move the expression to the left-hand side and change its sign   8x4+20x3+28x2+20x+8−9x4−27x2−9−18x3−18x=0modified 8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 9 x to the fourth power minus 27 x squared minus 9 minus 18 x cubed minus 18 x with boxed outline equals modified 0 with boxed outline8𝑥4+20𝑥3+28𝑥2+20𝑥+8−9𝑥4−27𝑥2−9−18𝑥3−18𝑥=0     Step   8x4+20x3+28x2+20x+8−9x4−27x2−9−18x3−18x=08 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 9 x to the fourth power minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 08𝑥4+20𝑥3+28𝑥2+20𝑥+8−9𝑥4−27𝑥2−9−18𝑥3−18𝑥=0       8x4+20x3+28x2+20x+8−9x4−27x2−9−18x3−18x=0modified 8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 9 x to the fourth power with boxed outline minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 08𝑥4+20𝑥3+28𝑥2+20𝑥+8−9𝑥4−27𝑥2−9−18𝑥3−18𝑥=0   Collect like terms   −x4+20x3+28x2+20x+8−27x2−9−18x3−18x=0modified negative x to the fourth power with boxed outline plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 0−𝑥4+20𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥3−18𝑥=0     Step   −x4+20x3+28x2+20x+8−27x2−9−18x3−18x=0negative x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 0−𝑥4+20𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥3−18𝑥=0       −x4+20x3+28x2+20x+8−27x2−9−18x3−18x=0modified negative x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x cubed with boxed outline minus 18 x equals 0−𝑥4+20𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥3−18𝑥=0   Collect like terms   −x4+2x3+28x2+20x+8−27x2−9−18x=0negative x to the fourth power plus modified 2 x cubed with boxed outline plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x equals 0−𝑥4+2𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥=0     Step   −x4+2x3+28x2+20x+8−27x2−9−18x=0negative x to the fourth power plus 2 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x equals 0−𝑥4+2𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥=0       −x4+2x3+28x2+20x+8−27x2−9−18x=0modified negative x to the fourth power plus 2 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared with boxed outline minus 9 minus 18 x equals 0−𝑥4+2𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥=0   Collect like terms   −x4+2x3+x2+20x+8−9−18x=0negative x to the fourth power plus 2 x cubed plus modified x squared with boxed outline plus 20 x plus 8 minus 9 minus 18 x equals 0−𝑥4+2𝑥3+𝑥2+20𝑥+8−9−18𝑥=0     Step   −x4+2x3+x2+20x+8−9−18x=0negative x to the fourth power plus 2 x cubed plus x squared plus 20 x plus 8 minus 9 minus 18 x equals 0−𝑥4+2𝑥3+𝑥2+20𝑥+8−9−18𝑥=0       −x4+2x3+x2+20x+8−9−18x=0modified negative x to the fourth power plus 2 x cubed plus x squared plus 20 x plus 8 minus 9 minus 18 x with boxed outline equals 0−𝑥4+2𝑥3+𝑥2+20𝑥+8−9−18𝑥=0   Collect like terms   −x4+2x3+x2+2x+8−9=0negative x to the fourth power plus 2 x cubed plus x squared plus modified 2 x with boxed outline plus 8 minus 9 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥+8−9=0     Step   −x4+2x3+x2+2x+8−9=0negative x to the fourth power plus 2 x cubed plus x squared plus 2 x plus 8 minus 9 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥+8−9=0       −x4+2x3+x2+2x+8−9=0modified negative x to the fourth power plus 2 x cubed plus x squared plus 2 x plus 8 minus 9 with boxed outline equals 0−𝑥4+2𝑥3+𝑥2+2𝑥+8−9=0   Calculate the difference   −x4+2x3+x2+2x-1=0negative x to the fourth power plus 2 x cubed plus x squared plus 2 x modified negative 1 with boxed outline equals 0−𝑥4+2𝑥3+𝑥2+2𝑥−1=0     Step   −x4+2x3+x2+2x−1=0negative x to the fourth power plus 2 x cubed plus x squared plus 2 x minus 1 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥−1=0       −x4+2x3+x2+2x−1=0negative x to the fourth power plus modified 2 x cubed with boxed outline plus x squared plus 2 x minus 1 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥−1=0   Write    2x32 x cubed2𝑥3as a difference   −x4+3x3−x3+x2+2x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed with boxed outline plus x squared plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3+𝑥2+2𝑥−1=0     Step   −x4+3x3−x3+x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed plus x squared plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3+𝑥2+2𝑥−1=0       −x4+3x3−x3+x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed modified positive x squared with boxed outline plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3+𝑥2+2𝑥−1=0   Write    x2x squared𝑥2as a sum or difference   −x4+3x3−x3−x2+3x2−x2+2x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared with boxed outline plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2+2𝑥−1=0     Step   −x4+3x3−x3−x2+3x2−x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2+2𝑥−1=0       −x4+3x3−x3−x2+3x2−x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared modified positive 2 x with boxed outline minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2+2𝑥−1=0   Write    2x2 x2𝑥as a sum   −x4+3x3−x3−x2+3x2−x2−x+3x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared minus x plus 3 x with boxed outline minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2−𝑥+3𝑥−1=0     Step   −x4+3x3−x3−x2+3x2−x2−x+3x−1=0negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2−𝑥+3𝑥−1=0       −x4+3x3−x3−x2+3x2−x2−x+3x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed minus x squared with boxed outline plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2−𝑥+3𝑥−1=0   Factor out    −x2negative x squared−𝑥2from the expression   −x2⋅(x2−3x+1)−x3+3x2−x2−x+3x−1=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren with boxed outline minus x cubed plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥3+3𝑥2−𝑥2−𝑥+3𝑥−1=0     Step   −x2⋅(x2−3x+1)−x3+3x2−x2−x+3x−1=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x cubed plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥3+3𝑥2−𝑥2−𝑥+3𝑥−1=0       −x2⋅(x2−3x+1)−x3+3x2−x2−x+3x−1=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x cubed plus 3 x squared minus x squared minus x with boxed outline plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥3+3𝑥2−𝑥2−𝑥+3𝑥−1=0   Factor out    −xnegative x−𝑥from the expression   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−x2+3x−1=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren modified negative x center dot open paren x squared minus 3 x plus 1 close paren with boxed outline minus x squared plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2+3𝑥−1=0     Step   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−x2+3x−1=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus x squared plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2+3𝑥−1=0       −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−x2+3x−1=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus x squared plus 3 x minus 1 with boxed outline equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2+3𝑥−1=0   Factor out the negative sign from the expression   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−(x2−3x+1)=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren modified negative open paren x squared minus 3 x plus 1 close paren with boxed outline equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2−3𝑥+1=0     Step   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−(x2−3x+1)=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus open paren x squared minus 3 x plus 1 close paren equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2−3𝑥+1=0       −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−(x2−3x+1)=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus open paren x squared minus 3 x plus 1 close paren with boxed outline equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2−3𝑥+1=0   Factor out    −(x2−3x+1)negative open paren x squared minus 3 x plus 1 close paren−𝑥2−3𝑥+1from the expression   −(x2−3x+1)⋅(x2+x+1)=0modified negative open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren with boxed outline equals 0−𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0     Step   −(x2−3x+1)⋅(x2+x+1)=0negative open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0−𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0       −(x2−3x+1)⋅(x2+x+1)=0modified negative open paren x squared minus 3 x plus 1 close paren with boxed outline center dot open paren x squared plus x plus 1 close paren equals 0−𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0   Change the signs on both sides of the equation   (x2−3x+1)⋅(x2+x+1)=0open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0     Step   (x2−3x+1)⋅(x2+x+1)=0open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0       (x2−3x+1)⋅(x2+x+1)=0modified open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0 with boxed outline𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0   When the product of factors equals    000, at least one factor is    000      Step   x2−3x+1=0x2+x+1=02 lines; Line 1: x squared minus 3 x plus 1 equals 0; Line 2: x squared plus x plus 1 equals 0 end-lines;𝑥2−3𝑥+1=0𝑥2+𝑥+1=0       x2−3x+1=0x2+x+1=02 lines; Line 1: modified x squared minus 3 x plus 1 equals 0 with boxed outline; Line 2: x squared plus x plus 1 equals 0 end-lines;𝑥2−3𝑥+1=0𝑥2+𝑥+1=0   Solve the equation for    xx𝑥   x=3+52x=3−52x2+x+1=03 lines; Line 1: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 3: x squared plus x plus 1 equals 0 end-lines;𝑥=3+5√2𝑥=3−5√2𝑥2+𝑥+1=0     Step   x=3+52x=3−52x2+x+1=03 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: x squared plus x plus 1 equals 0 end-lines;𝑥=3+5√2𝑥=3−5√2𝑥2+𝑥+1=0       x=3+52x=3−52x2+x+1=03 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: modified x squared plus x plus 1 equals 0 with boxed outline end-lines;𝑥=3+5√2𝑥=3−5√2𝑥2+𝑥+1=0   Solve the equation for    xx𝑥   x=3+52x=3−52x∉R3 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: modified x is not an element of the real numbers with boxed outline end-lines;𝑥=3+5√2𝑥=3−5√2𝑥∉ℝ     Step   x=3+52x=3−52x∉R3 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: x is not an element of the real numbers end-lines;𝑥=3+5√2𝑥=3−5√2𝑥∉ℝ       x=3+52x=3−52x∉R3 lines; Line 1: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 3: modified x is not an element of the real numbers with boxed outline end-lines;𝑥=3+5√2𝑥=3−5√2𝑥∉ℝ   Find the union   x=3−52x=3+522 lines; Line 1: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;𝑥=3−5√2𝑥=3+5√2     Step   x=3−52x=3+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√2𝑥=3+5√2       x=3−52x=3+522 lines; Line 1: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√2𝑥=3+5√2   Check if the given value is the solution of the equation   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−52x=3+522 lines; Line 1: modified the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√2𝑥=3+5√2     Step   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−52x=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√2𝑥=3+5√2       2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−52x=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√2𝑥=3+5√2   Check if the given value is the solution of the equation   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−522⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: modified the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√22⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2     Step   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−522⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√22⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2       2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−522⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: modified the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root with boxed outline equals modified the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√22⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2   Simplify the expression   5−12=5−122⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: modified the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction with boxed outline equals modified the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction with boxed outline; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;5√−12=5√−122⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2     Step   5−12=5−122⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;5√−12=5√−122⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2       5−12=5−122⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: modified the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root with boxed outline equals modified the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline end-lines;5√−12=5√−122⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2   Simplify the expression   5−12=5−121+52=1+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: modified the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline equals modified the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;5√−12=5√−121+5√2=1+5√2     Step   5−12=5−121+52=1+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;5√−12=5√−121+5√2=1+5√2       5−12=5−121+52=1+522 lines; Line 1: modified the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction with boxed outline; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;5√−12=5√−121+5√2=1+5√2   The equality is true, therefore    x=3−52x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction𝑥=3−5√2is a solution of the equation   x=3−521+52=1+522 lines; Line 1: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√21+5√2=1+5√2     Step   x=3−521+52=1+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√21+5√2=1+5√2       x=3−521+52=1+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: modified the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;𝑥=3−5√21+5√2=1+5√2   The equality is true, therefore    x=3+52x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction𝑥=3+5√2is a solution of the equation   x=3−52x=3+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;𝑥=3−5√2𝑥=3+5√2     Step   x=3−52x=3+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√2𝑥=3+5√2        The equation has    222solutions, so we'll label them as    x1x sub 1𝑥1and    x2x sub 2𝑥2   x1=3−52,x2=3+52modified x sub 1 equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction comma x sub 2 equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline𝑥1=3−5√2,𝑥2=3+5√2   Solution    x1=3−52,x2=3+52x sub 1 equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction comma x sub 2 equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction𝑥1=3−5√2,𝑥2=3+5√2 Alternative form     x1≈0.381966x sub 1 is approximately equal to 0.381966𝑥1≈0.381966,Câu trả lời: Step   2x2+3x+2−x2+x+1=xthe square root of 2 x squared plus 3 x plus 2 end-root minus the square root of x squared plus x plus 1 end-root equals the square root of x end-root2𝑥2+3𝑥+2√−𝑥2+𝑥+1√=𝑥√        2x2+3x+2−x2+x+1=xmodified the square root of 2 x squared plus 3 x plus 2 end-root minus the square root of x squared plus x plus 1 end-root equals the square root of x end-root with boxed outline2𝑥2+3𝑥+2√−𝑥2+𝑥+1√=𝑥√   Square both sides of the equation    2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+x+1=xmodified 2 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus x plus 1 equals x with boxed outline2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+𝑥+1=𝑥       Step   2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+x+1=x2 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus x plus 1 equals x2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+𝑥+1=𝑥       2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+x+1=x2 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared modified positive x with boxed outline plus 1 equals modified x with boxed outline2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+𝑥+1=𝑥   Cancel equal terms on both sides of the equation   2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus 1 equals modified 0 with boxed outline2𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+1=0     Step   2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+1=0       2x2+3x+2−2(2x2+3x+2)⋅(x2+x+1)+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of modified open paren 2 x squared plus 3 x plus 2 close paren center dot open paren x squared plus x plus 1 close paren with boxed outline end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥2+3𝑥+2⋅𝑥2+𝑥+1+𝑥2+1=0   Simplify the expression   2x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of modified 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 with boxed outline end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2+𝑥2+1=0     Step   2x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+x2+1=02 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus x squared plus 1 equals 02𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+𝑥2+1=0       2x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+x2+1=0modified 2 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus x squared with boxed outline plus 1 equals 02𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+𝑥2+1=0   Collect like terms   3x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+1=0modified 3 x squared with boxed outline plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus 1 equals 03𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+1=0     Step   3x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+1=03 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus 1 equals 03𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+1=0       3x2+3x+2−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2+1=0modified 3 x squared plus 3 x plus 2 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root plus 1 with boxed outline equals 03𝑥2+3𝑥+2−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√+1=0   Add the numbers   3x2+3x+3−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus modified 3 with boxed outline minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√=0     Step   3x2+3x+3−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2√=0       3x2+3x+3−22x4+2x3+2x2+3x3+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of modified 2 x to the fourth power plus 2 x cubed plus 2 x squared plus 3 x cubed with boxed outline plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+2𝑥3+2𝑥2+3𝑥3+3𝑥2+3𝑥+2𝑥2+2𝑥+2=0   Collect like terms   3x2+3x+3−22x4+5x3+2x2+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus modified 5 x cubed with boxed outline plus 2 x squared plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+2𝑥2+3𝑥2+3𝑥+2𝑥2+2𝑥+2=0     Step   3x2+3x+3−22x4+5x3+2x2+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 2 x squared plus 3 x squared plus 3 x plus 2 x squared plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+2𝑥2+3𝑥2+3𝑥+2𝑥2+2𝑥+2√=0       3x2+3x+3−22x4+5x3+2x2+3x2+3x+2x2+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of modified 2 x to the fourth power plus 5 x cubed plus 2 x squared plus 3 x squared plus 3 x plus 2 x squared with boxed outline plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+2𝑥2+3𝑥2+3𝑥+2𝑥2+2𝑥+2=0   Collect like terms   3x2+3x+3−22x4+5x3+7x2+3x+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus modified 7 x squared with boxed outline plus 3 x plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+3𝑥+2𝑥+2=0     Step   3x2+3x+3−22x4+5x3+7x2+3x+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 3 x plus 2 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+3𝑥+2𝑥+2√=0       3x2+3x+3−22x4+5x3+7x2+3x+2x+2=03 x squared plus 3 x plus 3 minus 2 the square root of modified 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 3 x plus 2 x with boxed outline plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+3𝑥+2𝑥+2=0   Collect like terms   3x2+3x+3−22x4+5x3+7x2+5x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus modified 5 x with boxed outline plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2=0     Step   3x2+3x+3−22x4+5x3+7x2+5x+2=03 x squared plus 3 x plus 3 minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals 03𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=0       3x2+3x+3−22x4+5x3+7x2+5x+2=0modified 3 x squared plus 3 x plus 3 with boxed outline minus 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals modified 0 with boxed outline3𝑥2+3𝑥+3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=0   Move the expression to the right-hand side and change its sign   -22x4+5x3+7x2+5x+2=-3x2−3x−3negative 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals modified negative 3 x squared minus 3 x minus 3 with boxed outline−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=−3𝑥2−3𝑥−3     Step   -22x4+5x3+7x2+5x+2=-3x2−3x−3negative 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals negative 3 x squared minus 3 x minus 3−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=−3𝑥2−3𝑥−3       -22x4+5x3+7x2+5x+2=-3x2−3x−3modified negative 2 the square root of 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 end-root equals negative 3 x squared minus 3 x minus 3 with boxed outline−22𝑥4+5𝑥3+7𝑥2+5𝑥+2√=−3𝑥2−3𝑥−3   Square both sides of the equation   4(2x4+5x3+7x2+5x+2)=9x4+9x2+9+18x3+18x2+18xmodified 4 open paren 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 close paren equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x with boxed outline42𝑥4+5𝑥3+7𝑥2+5𝑥+2=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥     Step   4(2x4+5x3+7x2+5x+2)=9x4+9x2+9+18x3+18x2+18x4 open paren 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 close paren equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x42𝑥4+5𝑥3+7𝑥2+5𝑥+2=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥       4(2x4+5x3+7x2+5x+2)=9x4+9x2+9+18x3+18x2+18xmodified 4 open paren 2 x to the fourth power plus 5 x cubed plus 7 x squared plus 5 x plus 2 close paren with boxed outline equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x42𝑥4+5𝑥3+7𝑥2+5𝑥+2=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥   Distribute    444through the parentheses   8x4+20x3+28x2+20x+8=9x4+9x2+9+18x3+18x2+18xmodified 8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 with boxed outline equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥     Step   8x4+20x3+28x2+20x+8=9x4+9x2+9+18x3+18x2+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥       8x4+20x3+28x2+20x+8=9x4+9x2+9+18x3+18x2+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals modified 9 x to the fourth power plus 9 x squared plus 9 plus 18 x cubed plus 18 x squared with boxed outline plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+9𝑥2+9+18𝑥3+18𝑥2+18𝑥   Collect like terms   8x4+20x3+28x2+20x+8=9x4+27x2+9+18x3+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals 9 x to the fourth power plus modified 27 x squared with boxed outline plus 9 plus 18 x cubed plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+27𝑥2+9+18𝑥3+18𝑥     Step   8x4+20x3+28x2+20x+8=9x4+27x2+9+18x3+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals 9 x to the fourth power plus 27 x squared plus 9 plus 18 x cubed plus 18 x8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+27𝑥2+9+18𝑥3+18𝑥       8x4+20x3+28x2+20x+8=9x4+27x2+9+18x3+18x8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 equals modified 9 x to the fourth power plus 27 x squared plus 9 plus 18 x cubed plus 18 x with boxed outline8𝑥4+20𝑥3+28𝑥2+20𝑥+8=9𝑥4+27𝑥2+9+18𝑥3+18𝑥   Move the expression to the left-hand side and change its sign   8x4+20x3+28x2+20x+8−9x4−27x2−9−18x3−18x=0modified 8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 9 x to the fourth power minus 27 x squared minus 9 minus 18 x cubed minus 18 x with boxed outline equals modified 0 with boxed outline8𝑥4+20𝑥3+28𝑥2+20𝑥+8−9𝑥4−27𝑥2−9−18𝑥3−18𝑥=0     Step   8x4+20x3+28x2+20x+8−9x4−27x2−9−18x3−18x=08 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 9 x to the fourth power minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 08𝑥4+20𝑥3+28𝑥2+20𝑥+8−9𝑥4−27𝑥2−9−18𝑥3−18𝑥=0       8x4+20x3+28x2+20x+8−9x4−27x2−9−18x3−18x=0modified 8 x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 9 x to the fourth power with boxed outline minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 08𝑥4+20𝑥3+28𝑥2+20𝑥+8−9𝑥4−27𝑥2−9−18𝑥3−18𝑥=0   Collect like terms   −x4+20x3+28x2+20x+8−27x2−9−18x3−18x=0modified negative x to the fourth power with boxed outline plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 0−𝑥4+20𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥3−18𝑥=0     Step   −x4+20x3+28x2+20x+8−27x2−9−18x3−18x=0negative x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x cubed minus 18 x equals 0−𝑥4+20𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥3−18𝑥=0       −x4+20x3+28x2+20x+8−27x2−9−18x3−18x=0modified negative x to the fourth power plus 20 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x cubed with boxed outline minus 18 x equals 0−𝑥4+20𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥3−18𝑥=0   Collect like terms   −x4+2x3+28x2+20x+8−27x2−9−18x=0negative x to the fourth power plus modified 2 x cubed with boxed outline plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x equals 0−𝑥4+2𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥=0     Step   −x4+2x3+28x2+20x+8−27x2−9−18x=0negative x to the fourth power plus 2 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared minus 9 minus 18 x equals 0−𝑥4+2𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥=0       −x4+2x3+28x2+20x+8−27x2−9−18x=0modified negative x to the fourth power plus 2 x cubed plus 28 x squared plus 20 x plus 8 minus 27 x squared with boxed outline minus 9 minus 18 x equals 0−𝑥4+2𝑥3+28𝑥2+20𝑥+8−27𝑥2−9−18𝑥=0   Collect like terms   −x4+2x3+x2+20x+8−9−18x=0negative x to the fourth power plus 2 x cubed plus modified x squared with boxed outline plus 20 x plus 8 minus 9 minus 18 x equals 0−𝑥4+2𝑥3+𝑥2+20𝑥+8−9−18𝑥=0     Step   −x4+2x3+x2+20x+8−9−18x=0negative x to the fourth power plus 2 x cubed plus x squared plus 20 x plus 8 minus 9 minus 18 x equals 0−𝑥4+2𝑥3+𝑥2+20𝑥+8−9−18𝑥=0       −x4+2x3+x2+20x+8−9−18x=0modified negative x to the fourth power plus 2 x cubed plus x squared plus 20 x plus 8 minus 9 minus 18 x with boxed outline equals 0−𝑥4+2𝑥3+𝑥2+20𝑥+8−9−18𝑥=0   Collect like terms   −x4+2x3+x2+2x+8−9=0negative x to the fourth power plus 2 x cubed plus x squared plus modified 2 x with boxed outline plus 8 minus 9 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥+8−9=0     Step   −x4+2x3+x2+2x+8−9=0negative x to the fourth power plus 2 x cubed plus x squared plus 2 x plus 8 minus 9 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥+8−9=0       −x4+2x3+x2+2x+8−9=0modified negative x to the fourth power plus 2 x cubed plus x squared plus 2 x plus 8 minus 9 with boxed outline equals 0−𝑥4+2𝑥3+𝑥2+2𝑥+8−9=0   Calculate the difference   −x4+2x3+x2+2x-1=0negative x to the fourth power plus 2 x cubed plus x squared plus 2 x modified negative 1 with boxed outline equals 0−𝑥4+2𝑥3+𝑥2+2𝑥−1=0     Step   −x4+2x3+x2+2x−1=0negative x to the fourth power plus 2 x cubed plus x squared plus 2 x minus 1 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥−1=0       −x4+2x3+x2+2x−1=0negative x to the fourth power plus modified 2 x cubed with boxed outline plus x squared plus 2 x minus 1 equals 0−𝑥4+2𝑥3+𝑥2+2𝑥−1=0   Write    2x32 x cubed2𝑥3as a difference   −x4+3x3−x3+x2+2x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed with boxed outline plus x squared plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3+𝑥2+2𝑥−1=0     Step   −x4+3x3−x3+x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed plus x squared plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3+𝑥2+2𝑥−1=0       −x4+3x3−x3+x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed modified positive x squared with boxed outline plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3+𝑥2+2𝑥−1=0   Write    x2x squared𝑥2as a sum or difference   −x4+3x3−x3−x2+3x2−x2+2x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared with boxed outline plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2+2𝑥−1=0     Step   −x4+3x3−x3−x2+3x2−x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared plus 2 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2+2𝑥−1=0       −x4+3x3−x3−x2+3x2−x2+2x−1=0negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared modified positive 2 x with boxed outline minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2+2𝑥−1=0   Write    2x2 x2𝑥as a sum   −x4+3x3−x3−x2+3x2−x2−x+3x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared minus x plus 3 x with boxed outline minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2−𝑥+3𝑥−1=0     Step   −x4+3x3−x3−x2+3x2−x2−x+3x−1=0negative x to the fourth power plus 3 x cubed minus x cubed minus x squared plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2−𝑥+3𝑥−1=0       −x4+3x3−x3−x2+3x2−x2−x+3x−1=0modified negative x to the fourth power plus 3 x cubed minus x cubed minus x squared with boxed outline plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥4+3𝑥3−𝑥3−𝑥2+3𝑥2−𝑥2−𝑥+3𝑥−1=0   Factor out    −x2negative x squared−𝑥2from the expression   −x2⋅(x2−3x+1)−x3+3x2−x2−x+3x−1=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren with boxed outline minus x cubed plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥3+3𝑥2−𝑥2−𝑥+3𝑥−1=0     Step   −x2⋅(x2−3x+1)−x3+3x2−x2−x+3x−1=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x cubed plus 3 x squared minus x squared minus x plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥3+3𝑥2−𝑥2−𝑥+3𝑥−1=0       −x2⋅(x2−3x+1)−x3+3x2−x2−x+3x−1=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x cubed plus 3 x squared minus x squared minus x with boxed outline plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥3+3𝑥2−𝑥2−𝑥+3𝑥−1=0   Factor out    −xnegative x−𝑥from the expression   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−x2+3x−1=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren modified negative x center dot open paren x squared minus 3 x plus 1 close paren with boxed outline minus x squared plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2+3𝑥−1=0     Step   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−x2+3x−1=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus x squared plus 3 x minus 1 equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2+3𝑥−1=0       −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−x2+3x−1=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus x squared plus 3 x minus 1 with boxed outline equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2+3𝑥−1=0   Factor out the negative sign from the expression   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−(x2−3x+1)=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren modified negative open paren x squared minus 3 x plus 1 close paren with boxed outline equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2−3𝑥+1=0     Step   −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−(x2−3x+1)=0negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus open paren x squared minus 3 x plus 1 close paren equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2−3𝑥+1=0       −x2⋅(x2−3x+1)−x⋅(x2−3x+1)−(x2−3x+1)=0modified negative x squared center dot open paren x squared minus 3 x plus 1 close paren minus x center dot open paren x squared minus 3 x plus 1 close paren minus open paren x squared minus 3 x plus 1 close paren with boxed outline equals 0−𝑥2⋅𝑥2−3𝑥+1−𝑥⋅𝑥2−3𝑥+1−𝑥2−3𝑥+1=0   Factor out    −(x2−3x+1)negative open paren x squared minus 3 x plus 1 close paren−𝑥2−3𝑥+1from the expression   −(x2−3x+1)⋅(x2+x+1)=0modified negative open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren with boxed outline equals 0−𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0     Step   −(x2−3x+1)⋅(x2+x+1)=0negative open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0−𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0       −(x2−3x+1)⋅(x2+x+1)=0modified negative open paren x squared minus 3 x plus 1 close paren with boxed outline center dot open paren x squared plus x plus 1 close paren equals 0−𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0   Change the signs on both sides of the equation   (x2−3x+1)⋅(x2+x+1)=0open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0     Step   (x2−3x+1)⋅(x2+x+1)=0open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0       (x2−3x+1)⋅(x2+x+1)=0modified open paren x squared minus 3 x plus 1 close paren center dot open paren x squared plus x plus 1 close paren equals 0 with boxed outline𝑥2−3𝑥+1⋅𝑥2+𝑥+1=0   When the product of factors equals    000, at least one factor is    000      Step   x2−3x+1=0x2+x+1=02 lines; Line 1: x squared minus 3 x plus 1 equals 0; Line 2: x squared plus x plus 1 equals 0 end-lines;𝑥2−3𝑥+1=0𝑥2+𝑥+1=0       x2−3x+1=0x2+x+1=02 lines; Line 1: modified x squared minus 3 x plus 1 equals 0 with boxed outline; Line 2: x squared plus x plus 1 equals 0 end-lines;𝑥2−3𝑥+1=0𝑥2+𝑥+1=0   Solve the equation for    xx𝑥   x=3+52x=3−52x2+x+1=03 lines; Line 1: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 3: x squared plus x plus 1 equals 0 end-lines;𝑥=3+5√2𝑥=3−5√2𝑥2+𝑥+1=0     Step   x=3+52x=3−52x2+x+1=03 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: x squared plus x plus 1 equals 0 end-lines;𝑥=3+5√2𝑥=3−5√2𝑥2+𝑥+1=0       x=3+52x=3−52x2+x+1=03 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: modified x squared plus x plus 1 equals 0 with boxed outline end-lines;𝑥=3+5√2𝑥=3−5√2𝑥2+𝑥+1=0   Solve the equation for    xx𝑥   x=3+52x=3−52x∉R3 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: modified x is not an element of the real numbers with boxed outline end-lines;𝑥=3+5√2𝑥=3−5√2𝑥∉ℝ     Step   x=3+52x=3−52x∉R3 lines; Line 1: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 3: x is not an element of the real numbers end-lines;𝑥=3+5√2𝑥=3−5√2𝑥∉ℝ       x=3+52x=3−52x∉R3 lines; Line 1: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 3: modified x is not an element of the real numbers with boxed outline end-lines;𝑥=3+5√2𝑥=3−5√2𝑥∉ℝ   Find the union   x=3−52x=3+522 lines; Line 1: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;𝑥=3−5√2𝑥=3+5√2     Step   x=3−52x=3+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√2𝑥=3+5√2       x=3−52x=3+522 lines; Line 1: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√2𝑥=3+5√2   Check if the given value is the solution of the equation   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−52x=3+522 lines; Line 1: modified the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√2𝑥=3+5√2     Step   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−52x=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√2𝑥=3+5√2       2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−52x=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√2𝑥=3+5√2   Check if the given value is the solution of the equation   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−522⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: modified the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√22⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2     Step   2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−522⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√22⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2       2⋅(3−52)2+3⋅3−52+2−(3−52)2+3−52+1=3−522⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: modified the square root of 2 center dot open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root with boxed outline equals modified the square root of the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;2⋅3−5√22+3⋅3−5√2+2⎷−3−5√22+3−5√2+1⎷=3−5√22⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2   Simplify the expression   5−12=5−122⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: modified the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction with boxed outline equals modified the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction with boxed outline; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;5√−12=5√−122⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2     Step   5−12=5−122⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root equals the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root end-lines;5√−12=5√−122⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2       5−12=5−122⋅(3+52)2+3⋅3+52+2−(3+52)2+3+52+1=3+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: modified the square root of 2 center dot open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus 3 center dot the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 2 end-root minus the square root of open paren the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction close paren squared plus the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction plus 1 end-root with boxed outline equals modified the square root of the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-root with boxed outline end-lines;5√−12=5√−122⋅3+5√22+3⋅3+5√2+2⎷−3+5√22+3+5√2+1⎷=3+5√2   Simplify the expression   5−12=5−121+52=1+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: modified the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline equals modified the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;5√−12=5√−121+5√2=1+5√2     Step   5−12=5−121+52=1+522 lines; Line 1: the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;5√−12=5√−121+5√2=1+5√2       5−12=5−121+52=1+522 lines; Line 1: modified the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction equals the fraction with numerator the square root of 5 end-root minus 1 and denominator 2 end-fraction with boxed outline; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;5√−12=5√−121+5√2=1+5√2   The equality is true, therefore    x=3−52x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction𝑥=3−5√2is a solution of the equation   x=3−521+52=1+522 lines; Line 1: modified x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction with boxed outline; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√21+5√2=1+5√2     Step   x=3−521+52=1+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√21+5√2=1+5√2       x=3−521+52=1+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: modified the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;𝑥=3−5√21+5√2=1+5√2   The equality is true, therefore    x=3+52x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction𝑥=3+5√2is a solution of the equation   x=3−52x=3+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: modified x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline end-lines;𝑥=3−5√2𝑥=3+5√2     Step   x=3−52x=3+522 lines; Line 1: x equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction; Line 2: x equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction end-lines;𝑥=3−5√2𝑥=3+5√2        The equation has    222solutions, so we'll label them as    x1x sub 1𝑥1and    x2x sub 2𝑥2   x1=3−52,x2=3+52modified x sub 1 equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction comma x sub 2 equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction with boxed outline𝑥1=3−5√2,𝑥2=3+5√2   Solution    x1=3−52,x2=3+52x sub 1 equals the fraction with numerator 3 minus the square root of 5 end-root and denominator 2 end-fraction comma x sub 2 equals the fraction with numerator 3 plus the square root of 5 end-root and denominator 2 end-fraction𝑥1=3−5√2,𝑥2=3+5√2 Alternative form     x1≈0.381966x sub 1 is approximately equal to 0.381966𝑥1≈0.381966,

ºvº☞†®üñg đẹp †®åî · Answer

Chúng ta cần giải phương trình:$\frac{2 x^{2} + 3 x + 2}{x^{2} + x + 1} = x$Bước 1: Nhân chéo để khử mẫuGiả sử $x^{2} + x + 1 \neq 0$, ta nhân hai vế với mẫu số $x^{2} + x + 1$:$2 x^{2} + 3 x + 2 = x \left(\right. x^{2} + x + 1 \left.\right)$Bước 2: Khai triển vế phải$x \left(\right. x^{2} + x + 1 \left.\right) = x^{3} + x^{2} + x$Giờ phương trình trở thành:$2 x^{2} + 3 x + 2 = x^{3} + x^{2} + x$Bước 3: Chuyển vế để giải phương trình$0 = x^{3} + x^{2} + x - 2 x^{2} - 3 x - 2$$0 = x^{3} - x^{2} - 2 x - 2$Bước 4: Giải phương trình bậc 3$x^{3} - x^{2} - 2 x - 2 = 0$Ta thử nghiệm nghiệm hữu tỉ bằng phân tích nhân tử hoặc thử nghiệm nghiệm:Thử $x = - 1$:$\left(\right. - 1 \left.\right)^{3} - \left(\right. - 1 \left.\right)^{2} - 2 \left(\right. - 1 \left.\right) - 2 = - 1 - 1 + 2 - 2 = - 2 \neq 0$Thử $x = 1$:$1 - 1 - 2 - 2 = - 4 \neq 0$Thử $x = - 2$:$\left(\right. - 2 \left.\right)^{3} - \left(\right. - 2 \left.\right)^{2} - 2 \left(\right. - 2 \left.\right) - 2 = - 8 - 4 + 4 - 2 = - 10 \neq 0$Thử $x = 2$:$8 - 4 - 4 - 2 = - 2 \neq 0$Không có nghiệm nguyên, ta dùng nhóm hạng tử:$x^{3} - x^{2} - 2 x - 2 = \left(\right. x^{2} \left.\right) \left(\right. x - 1 \left.\right) - 2 \left(\right. x + 1 \left.\right) = ?$Không tách được dễ, thử dùng phân tích đa thức bằng Horner:Dùng phương pháp Horner cho $f \left(\right. x \left.\right) = x^{3} - x^{2} - 2 x - 2$Dùng máy hoặc công cụ phân tích:Tìm được một nghiệm xấp xỉ: $x \approx 2.197$, nghiệm khác là nghiệm phức.Bước 5: Xét điều kiện xác địnhTa có mẫu số là $x^{2} + x + 1$Phương trình vô nghiệm khi:$x^{2} + x + 1 = 0$Giải: $\Delta = 1^{2} - 4 \left(\right. 1 \left.\right) \left(\right. 1 \left.\right) = - 3 < 0$→ Mẫu luôn khác 0, nên phương trình xác định với mọi x.Kết luận:Phương trình:$\frac{2 x^{2} + 3 x + 2}{x^{2} + x + 1} = x$tương đương:$x^{3} - x^{2} - 2 x - 2 = 0$Phương trình này không có nghiệm hữu tỉ, có 1 nghiệm thực duy nhất xấp xỉ:$x \approx 2.197$và 2 nghiệm phức.Câu trả lời: Chúng ta cần giải phương trình:$\frac{2 x^{2} + 3 x + 2}{x^{2} + x + 1} = x$Bước 1: Nhân chéo để khử mẫuGiả sử $x^{2} + x + 1 \neq 0$, ta nhân hai vế với mẫu số $x^{2} + x + 1$:$2 x^{2} + 3 x + 2 = x \left(\right. x^{2} + x + 1 \left.\right)$Bước 2: Khai triển vế phải$x \left(\right. x^{2} + x + 1 \left.\right) = x^{3} + x^{2} + x$Giờ phương trình trở thành:$2 x^{2} + 3 x + 2 = x^{3} + x^{2} + x$Bước 3: Chuyển vế để giải phương trình$0 = x^{3} + x^{2} + x - 2 x^{2} - 3 x - 2$$0 = x^{3} - x^{2} - 2 x - 2$Bước 4: Giải phương trình bậc 3$x^{3} - x^{2} - 2 x - 2 = 0$Ta thử nghiệm nghiệm hữu tỉ bằng phân tích nhân tử hoặc thử nghiệm nghiệm:Thử $x = - 1$:$\left(\right. - 1 \left.\right)^{3} - \left(\right. - 1 \left.\right)^{2} - 2 \left(\right. - 1 \left.\right) - 2 = - 1 - 1 + 2 - 2 = - 2 \neq 0$Thử $x = 1$:$1 - 1 - 2 - 2 = - 4 \neq 0$Thử $x = - 2$:$\left(\right. - 2 \left.\right)^{3} - \left(\right. - 2 \left.\right)^{2} - 2 \left(\right. - 2 \left.\right) - 2 = - 8 - 4 + 4 - 2 = - 10 \neq 0$Thử $x = 2$:$8 - 4 - 4 - 2 = - 2 \neq 0$Không có nghiệm nguyên, ta dùng nhóm hạng tử:$x^{3} - x^{2} - 2 x - 2 = \left(\right. x^{2} \left.\right) \left(\right. x - 1 \left.\right) - 2 \left(\right. x + 1 \left.\right) = ?$Không tách được dễ, thử dùng phân tích đa thức bằng Horner:Dùng phương pháp Horner cho $f \left(\right. x \left.\right) = x^{3} - x^{2} - 2 x - 2$Dùng máy hoặc công cụ phân tích:Tìm được một nghiệm xấp xỉ: $x \approx 2.197$, nghiệm khác là nghiệm phức.Bước 5: Xét điều kiện xác địnhTa có mẫu số là $x^{2} + x + 1$Phương trình vô nghiệm khi:$x^{2} + x + 1 = 0$Giải: $\Delta = 1^{2} - 4 \left(\right. 1 \left.\right) \left(\right. 1 \left.\right) = - 3 < 0$→ Mẫu luôn khác 0, nên phương trình xác định với mọi x.Kết luận:Phương trình:$\frac{2 x^{2} + 3 x + 2}{x^{2} + x + 1} = x$tương đương:$x^{3} - x^{2} - 2 x - 2 = 0$Phương trình này không có nghiệm hữu tỉ, có 1 nghiệm thực duy nhất xấp xỉ:$x \approx 2.197$và 2 nghiệm phức.

Khoá học trên OLM (olm.vn)

Khoá học trên OLM (olm.vn)