a. ĐKXĐ: $x\in\mathbb{R}$
PT \(\Rightarrow \left\{\begin{matrix} 2-x\geq 0\\ x^2+x+2=(3-x)^2\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x\leq 2\\ x^2+x+2=x^2-6x+9\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} x\leq 2\\ 7x=7\end{matrix}\right.\Leftrightarrow x=1\)
b. ĐKXĐ: $x\geq -1$
PT $\Leftrightarrow (x^2-1)+\sqrt{x+1}=0$
$\Leftrightarrow (x-1)(x+1)+\sqrt{x+1}=0$
$\Leftrightarrow \sqrt{x+1}[(x-1)\sqrt{x+1}+1]=0$
$\Leftrightarrow \sqrt{x+1}=0$ hoặc $(x-1)\sqrt{x+1}+1=0$
Nếu $\sqrt{x+1}=0$
$\Leftrightarrow x=-1$ (tm)
Nếu $(x-1)\sqrt{x+1}+1=0$
$\Leftrightarrow (x-1)\sqrt{x+1}=-1$
$\Rightarrow (x-1)^2(x+1)=1$
$\Leftrightarrow x^3-x^2-x=0$
$\Leftrightarrow x(x^2-x-1)=0$
$\Leftrightarrow x=0$ hoặc $x^2-x-1=0$
$\Leftrightarrow x=0$ hoặc $x=\frac{1\pm \sqrt{5}}{2}$
Kết hợp đkxđ suy ra $x=0; -1; \frac{1\pm \sqrt{5}}{2}$
c. ĐKXĐ: $x\geq 2$
PT $\Leftrightarrow \sqrt{(x-2)(x+2)}-2\sqrt{x-2}=0$
$\Leftrightarrow \sqrt{x-2}(\sqrt{x+2}-2)=0$
$\Leftrightarrow \sqrt{x-2}=0$ hoặc $\sqrt{x+2}-2=0$
$\Leftrightarrow x=2$ (thỏa mãn)
d. ĐKXĐ: $x\geq 3$ hoặc $x\leq -4$
PT \(\Rightarrow \left\{\begin{matrix} 8-x\geq 0\\ x^2+x-12=(8-x)^2\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x\leq 8\\ x^2+x-12=x^2-16x+64\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} x\leq 8\\ 17x=76\end{matrix}\right.\Leftrightarrow x=\frac{76}{17}\) (tm)
e. ĐKXĐ: $x\geq \frac{-3}{2}$
PT $\Leftrightarrow x=\sqrt{2x+3}$
\(\Rightarrow \left\{\begin{matrix} x\geq 0\\ x^2=2x+3\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x\geq 0\\ (x-3)(x+1)=0\end{matrix}\right.\Leftrightarrow x=3\) (tm)
f. ĐKXĐ: $x\geq 0$
PT $\Leftrightarrow \sqrt{x+2}=\sqrt{x}+\sqrt{x+1}$
$\Leftrightarrow x+2=2x+1+2\sqrt{x(x+1)}$ (bp hai vế)
$\Leftrightarrow 1-x=2\sqrt{x(x+1)}$
\(\Rightarrow \left\{\begin{matrix} 1-x\geq 0\\ (1-x)^2=4x(x+1)\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x\leq 1\\ 3x^2+6x-1=0\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} x\leq 1\\ x=\frac{-3\pm 2\sqrt{3}}{3}\end{matrix}\right.\)
Kết hợp cả đkxđ suy ra $x=\frac{-3+2\sqrt{3}}{3}$
