\(\sqrt{3x-2}+\sqrt{3+x}=\sqrt{5x+4}\)

→ \(\left(\sqrt{3x-2}+\sqrt{3+x}\right)^2=\left(\sqrt{5x+4}\right)^2\)

→ \(3x-2+3+x+2\sqrt{\left(2x-2\right)\left(3+x\right)}=5x+4\)

➝ \(4x+3+2\sqrt{6x+2x^2-6-2x}=5x+4\)

→ \(2\sqrt{2x^2+4x-6}=5x+4-4x-3\)

→ \(2\sqrt{2x^2+4x-6}=x+1\)

→ \(\left(2\sqrt{2x^2+4x-6}\right)^2=\left(x+1\right)^2\)

→ \(4\left(2x^2+4x-6\right)=x^2+2x+1\)

→ \(8x^2+16x-24=x^2+2x+1\)

→ \(8x^2+16x-24-x^2-2x-1=0\)

→ \(7x^2+14x-25=0\)

→ \(x_1=\frac{-7+4\sqrt{14}}{7}\)

\(x_2=\frac{-7-4\sqrt{14}}{7}\)

\( \sqrt {3x - 2} + \sqrt {3 + x} = \sqrt {5x + 4} \left( {x \ge \dfrac{2}{{3}}} \right)\\ \Leftrightarrow 3x - 2 + 2\sqrt {\left( {3x - 2} \right)\left( {3 + x} \right)} + 3 + x = 5x + 4\\ \Leftrightarrow 2\sqrt {7x + 3{x^2} - 6} = x + 3\\ \Leftrightarrow 4\left( {7x + 3{x^2} - 6} \right) = {x^2} + 6x + 9\\ \Leftrightarrow 28x + 12{x^2} - 24 = {x^2} + 6x + 9\\ \Leftrightarrow {x^2} + 2x - 3 = 0\\ \Leftrightarrow \left[ \begin{array}{l} x = 1\left( {tm} \right)\\ x = - 3\left( {ktm} \right) \end{array} \right. \)

ĐKXĐ : \(\left\{{}\begin{matrix}3x-2\ge0\\3+x\ge0\\5x+4\ge0\end{matrix}\right.\) => \(\left\{{}\begin{matrix}x\ge\frac{2}{3}\\x\ge-3\\x\ge-\frac{4}{5}\end{matrix}\right.\)

=> \(x\ge\frac{2}{3}\) (1)

Ta có : \(\sqrt{3x-2}+\sqrt{3+x}=\sqrt{5x+4}\)

<=> \(\left(\sqrt{3x-2}+\sqrt{3+x}\right)^2=\left(\sqrt{5x+4}\right)^2\)

<=> \(\left(3x-2\right)+2\sqrt{\left(3x-2\right)\left(3+x\right)}+\left(3+x\right)=5x+4\)

<=> \(3x-2+2\sqrt{\left(3x-2\right)\left(3+x\right)}+3+x=5x+4\)

<=> \(2\sqrt{\left(3x-2\right)\left(3+x\right)}=5x+4+2-3-x-3x\)

<=> \(2\sqrt{\left(3x-2\right)\left(3+x\right)}=x+3\)

<=> \(\sqrt{\left(3x-2\right)\left(3+x\right)}=\frac{x+3}{2}\)

ĐKXĐ : \(\frac{x+3}{2}\ge0\)

=> \(x+3\ge0\)

=> \(x\ge-3\) (2)

Từ (1) và (2)

=> \(x\ge\frac{2}{3}\)

<=> \(\left(\sqrt{\left(3x-2\right)\left(3+x\right)}\right)^2=\left(\frac{x+3}{2}\right)^2\)

<=> \(\left(3x-2\right)\left(3+x\right)=\frac{\left(x+3\right)^2}{4}\)

<=> \(9x-6+3x^2-2x=\frac{x^2+6x+9}{4}\)

<=> \(\frac{4\left(9x-6+3x^2-2x\right)}{4}=\frac{x^2+6x+9}{4}\)

<=> \(4\left(9x-6+3x^2-2x\right)=x^2+6x+9\)

<=> \(36x-24+12x^2-8x=x^2+6x+9\)

<=> \(36x-24+12x^2-8x-x^2-6x-9=0\)

<=> \(22x-33+11x^2=0\)

<=> \(11x^2+33x-11x-33=0\)

<=> \(11x\left(x-1\right)+33\left(x-1\right)=0\)

<=> \(\left(11x+33\right)\left(x-1\right)=0\)

<=> \(\left\{{}\begin{matrix}11x+33=0\\x-1=0\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}x=-3\left(L\right)\\x=1\left(TM\right)\end{matrix}\right.\)

Vậy phương trình trên có nghiệm là x = 1 .

X=7,3267

ĐKXĐ : \(\left\{{}\begin{matrix}2x+9\ge0\\4-x\ge0\\3x+1\ge0\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}2x\ge-9\\-x\ge-4\\3x\ge-1\end{matrix}\right.\) <=>\(\left\{{}\begin{matrix}x\ge-\frac{9}{2}\\x\le4\\x\ge-\frac{1}{3}\end{matrix}\right.\)

<=> \(4\ge x\ge-\frac{1}{3}\)

Ta có : \(\sqrt{2x+9}=\sqrt{4-x}+\sqrt{3x+1}\)

<=> \(\left(\sqrt{2x+9}\right)^2=\left(\sqrt{4-x}+\sqrt{3x+1}\right)^2\)

<=> \(2x+9=\left(4-x\right)+2\sqrt{\left(4-x\right)\left(3x+1\right)}+\left(3x+1\right)\)

<=> \(2x+9=4-x+2\sqrt{12x-3x^2+4-x}+3x+1\)

<=> \(2x+9-4+x-3x-1=2\sqrt{12x-3x^2+4-x}\)

<=> \(4=2\sqrt{12x-3x^2+4-x}\)

<=> \(4^2=\left(2\sqrt{12x-3x^2+4-x}\right)^2\)

<=> \(16=4\left(12x-3x^2+4-x\right)\)

<=> \(4=12x-3x^2+4-x\)

<=> \(0=12x-3x^2-x\)

<=> \(0=11x-3x^2\)

<=> \(0=x\left(11-3x\right)\)

<=> \(\left\{{}\begin{matrix}x=0\\11-3x=0\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}x=0\\-3x=-11\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}x=0\\x=\frac{11}{3}\end{matrix}\right.\) ( TM )

\(\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=5\)

\(\Leftrightarrow\sqrt{x-1-2.\sqrt{x-1}.2+4}+\sqrt{x-1-2.\sqrt{x-1}.3+9}=5\)

\(\Leftrightarrow\sqrt{\left(\sqrt{x-1}-2\right)^2}+\sqrt{\left(\sqrt{x-1}-3\right)^2}=5\)

\(\Leftrightarrow\left|\sqrt{x-1}-2\right|+\left|\sqrt{x-1}-3\right|\)=5

bạn giải tiếp nhé

Ta có:\(\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8+6\sqrt{x-1}}\)(ĐK: \(x\ge1\))

\(=\sqrt{\left(x-1\right)-2\sqrt{x-1}.2+4}+\sqrt{\left(x-1\right)+2\sqrt{x-1}.3+9}\)

\(=\sqrt{\left(\sqrt{x-1}-2\right)^2}+\sqrt{\left(\sqrt{x-1}+3\right)^2}\)

\(=\left|\sqrt{x-1}-2\right|+\left|\sqrt{x-1}+3\right|\)

Thay vào phương trình ta được:

\(\left|\sqrt{x-1}-2\right|+\left|\sqrt{x-1}+3\right|=5\)

\(\Leftrightarrow\left|\sqrt{x-1}-2\right|+\sqrt{x-1}+3=5\)(vì \(\sqrt{x-1}\ge0\Rightarrow\sqrt{x-1}+3>0\))

-TH: \(\sqrt{x-1}-2\ge0\Leftrightarrow\sqrt{x-1}\ge2\Leftrightarrow x-1\ge4\Leftrightarrow x\ge3\)thì ta có:

\(\sqrt{x-1}-2+\sqrt{x-1}+3=5\)

\(\Leftrightarrow2\sqrt{x-1}=4\)

\(\Leftrightarrow\sqrt{x-1}=2\)

\(\Leftrightarrow x-1=4\)

\(\Leftrightarrow x=5\)

-TH:\(\sqrt{x-1}-2< 0\Leftrightarrow x< 3\) thì ta có:

\(2-\sqrt{x-1}+\sqrt{x-1}+3=5\)

\(\Leftrightarrow5=5\)(luôn đúng \(\forall1\le x< 3\))

Vậy nghiệm của phương trình là \(1\le x< 3\) và \(x=5\)

Vũ Minh TuấnLê Thị Thục Hiền@Nk>↑@

\(\sqrt{x-2+\sqrt{2x-5}}+\sqrt{x+2+3\sqrt{2x-5}}=7\sqrt{2}\) (*) (đk : \(x\ge\frac{5}{2}\))

Đặt \(\sqrt{2x-5}=a\left(a\ge0\right)\)

=> 2x-5=a²

<=> \(x=\frac{a^2+5}{2}\)

Có \(\sqrt{\frac{a^2+5}{2}-2+a}+\sqrt{\frac{a^2+5}{2}+2+3a}=7\sqrt{2}\)

<=> \(\sqrt{\frac{a^2+5-4+2a}{2}}+\sqrt{\frac{a^2+5+4+6a}{2}}=7\sqrt{2}\)

<=>\(\sqrt{\frac{a^2+2a+1}{2}}+\sqrt{\frac{a^2+6a+9}{2}}=7\sqrt{2}\)

<=> \(\frac{\sqrt{\left(a+1\right)^2}}{\sqrt{2}}+\frac{\sqrt{\left(a+3\right)^2}}{\sqrt{2}}=7\sqrt{2}\)

<=> \(\left|a+1\right|+\left|a+3\right|=7\sqrt{2}.\sqrt{2}\)

<=> \(a+1+a+3=14\)(do a\(\ge\)0)

<=> \(2a=10\) <=> a=5(t/m)

<=> \(\sqrt{2x-5}=5\)

<=> \(2x-5=25\) <=> \(x=15\)(tm pt (*))

Vậy pt (*) có tập nghiệm \(S=\left\{15\right\}\)

ĐKXĐ : \(x-1\ge0\)

=> \(x\ge1\)

Ta có : \(\sqrt{x-2\sqrt{x-1}}+\sqrt{x+2\sqrt{x-1}}=5\)

<=> \(\sqrt{x-1-2\sqrt{x-1}+1}+\sqrt{x-1+2\sqrt{x-1}+1}=5\)

<=> \(\sqrt{\left(x-1\right)-2\sqrt{x-1}+1}+\sqrt{\left(x-1\right)+2\sqrt{x-1}+1}=5\)

<=> \(\sqrt{\left(\sqrt{x-1}-1\right)^2}+\sqrt{\left(\sqrt{x-1}+1\right)^2}=5\)

<=> \(|\sqrt{x-1}-1|+|\sqrt{x-1}+1|=5\)

<=> \(|\sqrt{x-1}-1|+\sqrt{x-1}+1=5\) ( 1 )

+, TH 1 : \(\sqrt{x-1}-1\ge0\) <=> \(x\ge2\) . Khi đó phương trình (1) được :

\(\sqrt{x-1}-1+\sqrt{x-1}+1=5\)

<=> \(2\sqrt{x-1}=5\)

<=> \(\sqrt{x-1}=2,5\)

<=> \(x-1=6,25\)

<=> \(x=7,25\) ( TM )

TH 2 : \(\sqrt{x-1}-1\le0\) <=> \(x\le2\) . Khi đó phương trình (1) được :

\(1-\sqrt{x-1}+\sqrt{x-1}+1=5\)

<=> \(2=5\) ( Vô lý )

Vậy phương trình trên có nghiệm duy nhất là x = 7,25 .