Question

\(\sqrt{8+\sqrt{x}}+\sqrt{5-\sqrt{x}}=5\) (giải phương trình bên)

Answer 1

\(ĐK:x\ge0\\ PT\Leftrightarrow\left(\sqrt{8+\sqrt{x}}-3\right)+\left(\sqrt{5-\sqrt{x}}-2\right)=0\\ \Leftrightarrow\dfrac{\sqrt{x}-1}{\sqrt{8+\sqrt{x}}+3}+\dfrac{-\sqrt{x}+1}{\sqrt{5-\sqrt{x}}+2}=0\\ \Leftrightarrow\left(\sqrt{x}-1\right)\left(\dfrac{1}{\sqrt{8+\sqrt{x}}+3}-\dfrac{1}{\sqrt{5-\sqrt{x}}+2}\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=1\left(tm\right)\\\dfrac{1}{\sqrt{8+\sqrt{x}}+3}-\dfrac{1}{\sqrt{5-\sqrt{x}}+2}=0\left(vô.n_0,\forall x\ge0\right)\end{matrix}\right.\)

Vậy PT có nghiệm duy nhất \(x=1\)