Question

Please solve this.
Need all the steps.

Answer 1

please refer to attachment

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

We are aware of the Componendo - Dividendo rule that

$\implies \: \displaystyle \: \frac{a}{b} = \frac{c}{d} \: \: \: implies \: \: \frac{a + b}{a - b} = \frac{c + d}{c - d}$

GIVEN

$\displaystyle \: \frac{ \sqrt{3 - x} + \sqrt{3 + x} }{ \sqrt{3 - x} - \sqrt{3 + x}} = 5$

TO DETERMINE

The value of x

CALCULATION

$\displaystyle \: \frac{ \sqrt{3 - x} + \sqrt{3 + x} }{ \sqrt{3 - x} - \sqrt{3 + x}} = 5$

By the Componendo - Dividendo rule we get

$\implies \: \displaystyle \: \frac{ (\sqrt{3 - x} + \sqrt{3 + x}) + (\sqrt{3 - x} - \sqrt{3 + x} )\: }{ (\sqrt{3 - x} - \sqrt{3 + x}) - (\sqrt{3 - x} - \sqrt{3 + x})} = \frac{5 + 1}{5 - 1}$

$\implies \: \displaystyle \: \frac{ 2\sqrt{3 - x} }{ 2 \sqrt{3 + x}} = \frac{6}{4}$

$\implies \: \displaystyle \: \frac{ \sqrt{3 - x} }{ \sqrt{3 + x}} = \frac{3}{2}$

Squaring both sides

$\implies \: \displaystyle \: \frac{ {3 - x} }{ {3 + x}} = \frac{9}{4}$

$\implies \: 27 + 9x = 12 - 4x$

$\implies \: 13x = - 15$

$\displaystyle \: \therefore \: x = - \frac{15}{13}$