a) Ta thấy \(xy=\dfrac{\left(x+y\right)^2-\left(x^2+y^2\right)}{2}=\dfrac{3^2-5}{2}=2\)

\(\Rightarrow x^3+y^3=\left(x+y\right)\left(x^2+y^2-xy\right)\) \(=3\left(5-2\right)=9\)

 b) Ta thấy \(xy=\dfrac{-\left(x-y\right)^2+\left(x^2+y^2\right)}{2}=\dfrac{15-5^2}{2}=-5\)

\(\Rightarrow x^3-y^3=\left(x-y\right)\left(x^2+y^2+xy\right)\) \(=5\left(15-5\right)=50\)

1) 

Ta có: x+y=2

nên \(\left(x+y\right)^2=4\)

\(\Leftrightarrow x^2+y^2+2xy=4\)

\(\Leftrightarrow2xy=2\)

hay xy=1

Ta có: \(x^3+y^3\)

\(=\left(x+y\right)^3-3xy\left(x+y\right)\)

\(=2^3-3\cdot1\cdot2\)

=2

2)\(x^2+y^2=\left(x+y\right)^2-2xy=8^2-2\cdot\left(-20\right)=104\)

\(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=8^3-3\cdot\left(-20\right)\cdot8=512+480=992\)

\(x^2+y^2+xy=\left(x+y\right)^2-xy=8^2-\left(-20\right)=64+20=84\)

Lời giải:
a.

$x^3+y^3=(x+y)^3-3xy(x+y)=9^3-3.9.18=243$

$x^4+y^4=(x^2+y^2)^2-2x^2y^2=[(x+y)^2-2xy]^2-2x^2y^2$

$=[9^2-2.18]^2-2.18^2=1377$

Nếu $x\geq y$ thì:

$x^3-y^3=(x-y)(x^2+xy+y^2)$

$=|x-y|[(x+y)^2-xy]=\sqrt{(x+y)^2-4xy}[(x+y)^2-xy]$

$=\sqrt{9^2-4.18}(9^2-18)=189$

Nếu $x< y$ thì $x^3-y^3=-189$

b.

$A=(x+y)^2-6(x+y)+y-5$

$=(-9)^2-6(-9)+y-5=130+y$

Chưa đủ cơ sở để tính biểu thức.

a) \(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=9^3-3\cdot18\cdot9=243\)

\(x^4+y^4=\left(x^2+y^2\right)^2-2x^2y^2\)

\(=\left[\left(x+y\right)^2-2xy\right]^2-2\left(xy\right)^2\)

\(=\left(9^2-2\cdot18\right)^2-2\cdot18^2\)

\(=45^2-2\cdot324\)

=1377

`#3107.101107`

`D = x^3 - y^3 - 3xy` biết `x - y - 1 = 0`

Ta có:

`x - y - 1 = 0`

`=> x - y = 1`

`D = x^3 - y^3 - 3xy`

`= (x - y)(x^2 + xy + y^2) - 3xy`

`= 1 * (x^2 + xy + y^2) - 3xy`

`= x^2+ xy + y^2 - 3xy`

`= x^2 - 2xy + y^2`

`= x^2 - 2*x*y + y^2`

`= (x - y)^2`

`= 1^2 = 1`

Vậy, với `x - y = 1` thì `D = 1`

________

`E = x^3 + y^3` với `x + y = 5; x^2 + y^2 = 17`

`x + y = 5`

`=> (x + y)^2 = 25`

`=> x^2 + 2xy + y^2 = 25`

`=> 2xy = 25 - (x^2 + y^2)`

`=> 2xy = 25 - 17`

`=> 2xy = 8`

`=> xy = 4`

Ta có:

`E = x^3 + y^3`

`= (x + y)(x^2 - xy + y^2)`

`= 5 * [ (x^2 + y^2) - xy]`

`= 5 * (17 - 4)`

`= 5 * 13`

`= 65`

Vậy, với `x + y = 5; x^2 + y^2 = 17` thì `E = 65`

________

`F = x^3 - y^3` với `x - y = 4; x^2 + y^2 = 26`

Ta có:

`x - y = 4`

`=> (x - y)^2 = 16`

`=> x^2 - 2xy + y^2 = 16`

`=> (x^2 + y^2) - 2xy = 16`

`=> 2xy = (x^2 + y^2) - 16`

`=> 2xy = 26 - 16`

`=> 2xy = 10`

`=> xy = 5`

Ta có:

`F = x^3 - y^3`

`= (x - y)(x^2 + xy + y^2)`

`= 4 * [ (x^2 + y^2) + xy]`

`= 4 * (26 + 5)`

`= 4*31`

`= 124`

Vậy, với `x - y = 4; x^2 + y^2 = 26` thì `F = 124.`

a) \(\left(x-5\right)^2=\left(3+2x\right)^2\)

\(\Rightarrow\left(3+2x\right)^2-\left(x-5\right)^2=0\)

\(\Rightarrow\left(3+2x+x-5\right)\left(3+2x-x+5\right)=0\)

\(\Rightarrow\left(3x-2\right)\left(x+8\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}3x-2=0\\x+8=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\dfrac{2}{3}\\x=-8\end{matrix}\right.\)

b) \(27x^3-54x^2+36x=9\)

\(\Rightarrow27x^3-54x^2+36x-9=0\)

\(\Rightarrow27x^3-54x^2+36x-8+8-9=0\)

\(\Rightarrow\left(3x-2\right)^3-1=0\)

\(\Rightarrow\left(3x-2-1\right)\left[\left(3x-2\right)^2+3x-2+1\right]=0\)

\(\Rightarrow\left(3x-3\right)\left[\left(3x-2\right)^2+3x-2+\dfrac{1}{4}-\dfrac{1}{4}+1\right]=0\)

\(\Rightarrow\left(3x-3\right)\left[\left(3x-2+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\right]=0\)

\(\Rightarrow\left(3x-3\right)\left[\left(3x-\dfrac{3}{2}\right)^2+\dfrac{3}{4}\right]=0\left(1\right)\)

mà \(\left(3x-\dfrac{3}{2}\right)^2+\dfrac{3}{4}>0,\forall x\)

\(\left(1\right)\Rightarrow3x-3=0\Rightarrow3x=3\Rightarrow x=1\)

(\(x-5\))² = (3 +2\(x\))² ⇒ \(\left[{}\begin{matrix}x-5=3+2x\\x-5=-3-2x\end{matrix}\right.\) ⇒ \(\left[{}\begin{matrix}x=-8\\x=\dfrac{2}{3}\end{matrix}\right.\) vậy \(x\in\){-8; \(\dfrac{2}{3}\)}

  27\(x^3\) - 54\(x^2\) + 36\(x\) = 9

27\(x^3\) - 54\(x^2\) + 36\(x\) - 8 = 1

(3\(x\) - 2)³ = 1 ⇒ 3\(x\) - 2 = 1 ⇒ \(x\) = 1

1

\(a,x+y=1\Leftrightarrow\left(x+y\right)^3=1\Leftrightarrow x^3+y^3+3xy\left(x+y\right)=1\\ \Leftrightarrow x^3+y^3+3xy\cdot1=1\Leftrightarrow x^3+y^3+3xy=1\)

\(b,x^3-y^3-3xy\\ =x^3-3x^2y+3xy^2-y^3-3xy+3x^2y-3xy^2\\ =\left(x-y\right)^3-3xy\left(x-y-1\right)\\ =1^3-3xy\left(1-1\right)=1-0=1\)

\(c,x^3+y^3+3xy\left(x^2+y^2\right)+6x^2y^2\left(x+y\right)\\ =\left(x+y\right)\left(x^2-xy+y^2\right)+3xy\left[\left(x+y\right)^2-2xy\right]+6x^2y^2\\ =x^2-xy+y^2+3xy-6x^2y^2+6x^2y^2\\ =x^2+2xy+y^2=\left(x+y\right)^2=1\)

(x - 5)² = (3 + 2x)²

(x - 5)² - (3 + 2x)² = 0

[(x - 5) - (3 + 2x)][(x - 5) + (3 + 2x)] = 0

(x - 5 - 3 - 2x)(x - 5 + 3 + 2x) = 0

(-x - 8)(3x - 2) = 0

-x - 8 = 0 hoặc 3x - 2 = 0

*) -x - 8 = 0

-x = 8

x = -8

*) 3x - 2 = 0

3x = 2

x = 2/3

Vậy x = -8; x = 2/3

--------------------

27x³ - 54x² + 36x = 9

27x³ - 54x² + 36x - 9 = 0

27x³ - 27x² - 27x² + 27x + 9x - 9 = 0

(27x³ - 27x²) - (27x² - 27x) + (9x - 9) = 0

27x²(x - 1) - 27x(x - 1) + 9(x - 1) = 0

(x - 1)(27x² - 27x + 9) = 0

x - 1 = 0 hoặc 27x² - 27x + 9 = 0

*) x - 1 = 0

x = 1

*) 27x² - 27x + 9 = 0

Ta có:

27x² - 27x + 9

= 27(x² - x + 1/3)

= 27(x² - 2.x.1/2 + 1/4 + 1/12)

= 27[(x - 1/2)² + 1/12] > 0 với mọi x ∈ R

⇒ 27x² - 27x + 9 = 0 (vô lí)

Vậy x = 1

A = x² + y²

= x² - 2xy + y² + 2xy

= (x - y)² + 2xy

= 4² + 2.1

= 16 + 2

= 18

B = x³ - y³

= (x - y)(x² + xy + y²)

= (x - y)(x² - 2xy + y² + xy + 2xy)

= (x - y)[(x - y)² + 3xy]

= 4.(4² + 3.1)

= 4.(16 + 3)

= 4.19

= 76

C = x⁴ + y⁴

= (x²)² + (y²)²

= (x²)² + 2x²y² + (y²)² - 2x²y²

= (x² + y²)² - 2x²y²

= (x² - 2x²y² + y² + 2x²y²)² - 2x²y²

= [(x - y)² + 2x²y²]² - 2x²y²

= (4² + 2.1²)² - 2.1²

= (16 + 2)² - 2

= 18² - 2

= 324 - 2

= 322

a: =>(2x+3)^2-(x-5)^2=0

=>(2x+3+x-5)(2x+3-x+5)=0

=>(x+8)(3x-2)=0

=>x=2/3 hoặc x=-8

b: =>27x^3-54x^2-36x-9=0

=>3x^3-6x^2-4x-1=0

=>\(x\simeq2,57\)

c: A=x^2+y^2=(x-y)^2+2xy=4^2+2=18

B=x^3-y^3=(x-y)^3+3xy(x-y)

=4^3+3*1*4

=64+12=76

C=(x^2+y^2)^2-2x^2y^2

=18^2-2*1^2=322

Biến đổi tương đương nhé bạn.

a: Ta có: \(\left(x+y\right)^2\)

\(=x^2+2xy+y^2\)

\(\Leftrightarrow x^2+y^2=\dfrac{\left(x+y\right)^2}{2xy}\ge\dfrac{\left(x+y\right)^2}{2}\forall x,y>0\)

\(a,A=x^2+y^2\\=x^2-2xy+y^2+2xy\\=(x-y)^2+2xy\\=2^2+2\cdot1\\=4+2\\=6\)

\(b,x+y=1\\\Leftrightarrow (x+y)^3=1^3\\\Leftrightarrow x^3+3x^2y+3xy^2+y^3=1\\\Leftrightarrow x^3+3xy(x+y)+y^3=1\\\Leftrightarrow x^3+3xy\cdot1+y^3=1\\\Rightarrow A=1\)

a) Ta có:

\(x-y=2\)

\(\Rightarrow\left(x-y\right)^2=2^2\)

\(\Rightarrow x^2-2xy+y^2=4\)

Mà: \(xy=1\)

\(\Rightarrow\left(x^2+y^2\right)-2\cdot1=4\)

\(\Rightarrow x^2+y^2=4+2\)

\(\Rightarrow x^2+y^2=6\)

b) Ta có: 

\(x+y=1\)

\(\Rightarrow\left(x+y\right)^3=1^3\)

\(\Rightarrow x^3+3x^2y+3xy+y^3=1\)

\(\Rightarrow x^3+3xy\left(x+y\right)+y^3=1\) 

Mà: x + y = 1

\(\Rightarrow x^3+3xy\cdot1+y^3=1\)

\(\Rightarrow x^3+3xy+y^3=1\)

\(\left\{{}\begin{matrix}x-y=4\\xy=1\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x=y+4\\y\left(y+4\right)=1\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x=y+4\\y^2+4y-1=0\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x=y+4\\\left[{}\begin{matrix}y=-2+\sqrt{5}\\y=-2-\sqrt{5}\end{matrix}\right.\end{matrix}\right.\)

Với \(y=-2+\sqrt{5}\Rightarrow x=2+\sqrt{5}\)

Với \(y=-2-\sqrt{5}\Rightarrow x=2-\sqrt{5}\)

\(\Rightarrow A=x^2+y^2=\left(-2+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2=\left(2-\sqrt{5}\right)^2+\left(-2-\sqrt{5}\right)^2=18\)

\(B=x^3+y^3\Rightarrow\left[{}\begin{matrix}B=\left(2+\sqrt{5}\right)^3+\left(-2+\sqrt{5}\right)^3=34\sqrt{5}\\B=\left(2-\sqrt{5}\right)^3+\left(-2-\sqrt{5}\right)^3=-34\sqrt{5}\end{matrix}\right.\)

\(\Rightarrow C=x^4+y^4=\left(-2+\sqrt{5}\right)^4+\left(2+\sqrt{5}\right)^4=\left(2-\sqrt{5}\right)^4+\left(-2-\sqrt{5}\right)^4=322\)