Question

Решите уравнение √х+2 + √3-х=3

Answer 1

$\sqrt{x+2}+\sqrt{3-x}=3\Leftrightarrow \cfrac{\left ( \sqrt{x+2}+\sqrt{3-x} \right )\left ( \sqrt{x+2}-\sqrt{3-x} \right )}{\sqrt{x+2}-\sqrt{3-x}}=3\\\cfrac{2x-1}{\sqrt{x+2}-\sqrt{3-x}}=3\Leftrightarrow \sqrt{x+2}-\sqrt{3-x}=\cfrac{2x-1}{3}\\\begin{cases}\sqrt{x+2}+\sqrt{3-x}=3\\ \sqrt{x+2}-\sqrt{3-x}=\cfrac{2x-1}{3}\end{cases}\Rightarrow 3\sqrt{x+2}=x+4$$\begin{cases}9(x+2)=(x+4)^2\\x+4\geq 0\end{cases}\Leftrightarrow \begin{cases}9x+18=x^2+8x+16\\x\geq -4\end{cases}\Leftrightarrow \begin{cases}x^2-x-2=0\\x\geq -4\end{cases}\\\begin{cases}(x+1)(x-2)=0\\x\geq -4\end{cases}\Rightarrow \begin{cases}x=\left \{ -1,2 \right \}\\x\geq -4\end{cases}\Rightarrow x=\left \{ -1,2 \right \}$