Đặt \(\hept{\begin{cases}\sqrt{x-4}=a\\\sqrt{x+4}=b\end{cases}}\)

=> \(\hept{\begin{cases}a^2+b^2=2x\\b^2-a^2=8\\ab=\sqrt{x^2-16}\end{cases}}\)

Từ đó thì PT ban đầu thành

a + b = 2ab + a² + b² - 12

<=> (a + b)² - (a + b) - 12 = 0

<=> \(\hept{\begin{cases}\left(a+b\right)=4\\\left(a+b\right)=-3\left(loai\right)\end{cases}}\)

Tới đây thì đơn giản rồi bạn làm tiếp nhé

\(2\sqrt{2x+4}+4\sqrt{2-x}=\sqrt{9x^2+16}\)

\(\Leftrightarrow\left(2\sqrt{2x+4}+4\sqrt{2-x}\right)^2=\left(\sqrt{9x^2+16}\right)^2\)

\(\Leftrightarrow4\left(2x+4\right)+16\left(2-x\right)+16\sqrt{2x+4}\sqrt{2-x}=9x^2+16\)

\(\Leftrightarrow4.2\left(4-x^2\right)+16\sqrt{2\left(4-x^2\right)}=x^2+8x\)

Đặt \(\sqrt{2\left(4-x^2\right)}=a\)

\(\Rightarrow4a^2+16a=x^2+8x\)

\(\Leftrightarrow\left(2a-x\right)\left(2a+x+8\right)=0\)

Làm nốt

\(\sqrt{2x^2+16x+18}+\sqrt{x^2+1}=2x+4\left(1\right)\)

\(ĐK:x\in R\)

\(pt\left(1\right)\Leftrightarrow2x^2+16x+18+x^2+1+2\sqrt[]{(2x^2+16x+18)\left(x^2+1\right)}=4x^2+16x+16\)

\(\Leftrightarrow3+2\sqrt{(2x^2+16x+18)\left(x^2+1\right)}=x^2\)

\(\Leftrightarrow2\sqrt{(2x^2+16x+8)\left(x^2+1\right)}=x^2-3\)

\(\Leftrightarrow\left\{{}\begin{matrix}x^2-3\ge0\\4\left(2x^2+16x+8\right)\left(x^2+1\right)=x^4-6x^2+9\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-\sqrt{3}\le x\le\sqrt{3}\\4\left(2x^4+16x^3+10x^2+16x+8\right)=x^4-6x^2+9\end{matrix}\right.\)

\(\Leftrightarrow7x^4+64x^3+46x^2+64x+23=0\)

Không có ai trả lời thì cho mình vậy :))

\(\sqrt{x+4}\sqrt{x-4}=2x-12+2\sqrt{x^2-16}\)

\(\Rightarrow\sqrt{\left(x+4\right)\left(x-4\right)}=2x-12+2\sqrt{x^2-16}\)

\(\Leftrightarrow\sqrt{x^2-16}=2x-12+2\sqrt{x^2-16}\)

\(\Leftrightarrow\sqrt{x^2-16}-2\sqrt{x^2-16}=2x-12\)

\(\Leftrightarrow-\sqrt{x^2-16}=2x-12\)

\(\Leftrightarrow\sqrt{x^2-16}=-2x+12\)

\(\Leftrightarrow x^2-16=\left(-2x+12\right)^2\)

\(\Leftrightarrow x^2-16=4x^2-48x+144\)

\(\Leftrightarrow x^2-16-4x^2+48x-144=0\)

\(\Leftrightarrow-3x^2-160+48x=0\)

\(\Leftrightarrow-3x^2+48x-160=0\)

\(\Leftrightarrow3x^2-48x+160=0\)

\(\Leftrightarrow x=\dfrac{-\left(-48\right)\pm\sqrt{\left(-48\right)^2-4\cdot3\cdot160}}{2\cdot3}\)

\(\Leftrightarrow x=\dfrac{48\pm\sqrt{2304-1920}}{6}\)

\(\Leftrightarrow x=\dfrac{48\pm\sqrt{384}}{6}\)

\(\Leftrightarrow x=\dfrac{48+8\sqrt{6}}{6}\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{48+8\sqrt{6}}{6}\\x=\dfrac{48-8\sqrt{6}}{6}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{24+4\sqrt{6}}{3}\\x=\dfrac{24-4\sqrt{6}}{3}\end{matrix}\right.\)

Vậy \(x_1=\dfrac{24+4\sqrt{6}}{3};x_2=\dfrac{24-4\sqrt{6}}{3}\)