Find x from the following equations : 2-x+2+x 2-x-2+x =3 - Vidyadip

Question

Find $x$ from the following equations : $\frac{\sqrt{2-x}+\sqrt{2+x}}{\sqrt{2-x}-\sqrt{2+x}}=3$

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Answer

$
\frac{\sqrt{2-x}+\sqrt{2+x}}{\sqrt{2-x}-\sqrt{2+x}}=3
$Applying componendo and dividendo,
$
\begin{aligned}
& \frac{\sqrt{2-x}+\sqrt{2+x}+\sqrt{2-x}-\sqrt{2+x}}{\sqrt{2-x}+\sqrt{2+x}-\sqrt{2-x}+\sqrt{2+x}}=\frac{3+1}{3-1} \\
& \Rightarrow \frac{2 \sqrt{2-x}}{2 \sqrt{2+x}}=\frac{4}{2} \\
& \Rightarrow \frac{\sqrt{2-x}}{\sqrt{2+x}}=\frac{2}{1}
\end{aligned}$
Squaring both sides
$
\begin{aligned}
& \frac{2-x}{2+x}=\frac{4}{1} \\
& \Rightarrow 8+4 x=2-x \\
& 4 x+x=2 \\
& \Rightarrow 5 x=-6 \\
& \therefore x=\frac{-6}{5}
\end{aligned}
$

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