Question

Find the value of x by applying dividendo and componendo​

Answer 1

$\frac{\sqrt{x+5}+\sqrt{x-16} }{\sqrt{x+5}-\sqrt{x-16}} = \frac{7}{3}$

According to dividendo and componendo​ rule

If, $\frac{a}{b} = \frac{c}{d}$, then

$\big\black\boxed{\frac{a+b}{a-b}=\frac{c+d}{c-d} }$

Applying dividendo and componendo​ in given question

$\frac{\sqrt{x+5}+\sqrt{x-16} + \sqrt{x+5}-\sqrt{x-16} }{\sqrt{x+5}+\sqrt{x-16}-(\sqrt{x+5}-\sqrt{x-16})} = \frac{7+3}{7-3}$

$\frac{2\sqrt{x+5}}{2\sqrt{x-16}} = \frac{10}{4}$

$\frac{x+5}{x-16} = \frac{25}{4}$

Applying dividendo and componendo again

$\frac{x+5+x-16}{x+5-(x-16)} = \frac{25+4}{25-4}$

$\frac{2x-11}{21} = \frac{29}{21}$

$2x-11 = 29$

$2x = 40$

$\huge\black\boxed{x=20}$