Question

√x+13 +√x-1 / √x+13 - √x-1 = 3​

Answer 1

Step-by-step explanation:

GIVEN EQUATION : -

$\sf \frac{ \sqrt{x + 13} + \sqrt{x - 1} }{ \sqrt{x + 13} - \sqrt{x - 1} } = 3$

SOLUTION :-

$\sf \frac{ \sqrt{x + 13} + \sqrt{x - 1} }{ \sqrt{x + 13} - \sqrt{x - 1} } = \frac{3}{1}$

Use Componendo and Dividendo

$\sf \frac{ \sqrt{x + 13} + \sqrt{x - 1} }{ \sqrt{x + 13} - \sqrt{x - 1} } = \frac{3}{1} \\ \\ \implies \sf \frac{ (\sqrt{x + 13} + \sqrt{x - 1} ) + (\sqrt{x + 13} - \sqrt{x - 1} )}{ (\sqrt{x + 13} + \sqrt{x - 1} ) - (\sqrt{x + 13} - \sqrt{x - 1} )} = \frac{3 + 1}{3 - 1} \\ \\ \implies\sf \frac{ \sqrt{x + 13} + \sqrt{x - 1} + \sqrt{x + 13} - \sqrt{x - 1} }{ \sqrt{x + 13} + \sqrt {x - 1} - \sqrt{x + 13} + \sqrt{x - 1} } = \frac{4}{2} \\ \\ \implies \sf\frac{2 \sqrt{x + 13} }{2 \sqrt{x - 1} } = 2 \\ \\ \sf \implies \frac{ \sqrt{x + 13} }{ \sqrt{x - 1} } = 2 \\ \rm \: squaring \: \: both \: \: sides \\ \implies \sf \bigg( \sqrt{ \frac{x + 13}{x - 1} } \bigg) {}^{2} = {2}^{2} \\ \\ \implies \sf\frac{x + 13}{x - 1} = 4 \\ \\ \implies \sf \: x + 1 3= 4(x - 1) \\ \\ \implies \: \sf \: x + 13 = 4x - 4 \\ \\ \implies \: \sf \: x - 4x = - 4 - 13 \\ \\ \implies \: \sf - 3x = - 17 \\ \\ \implies \: \sf \: x \: = \frac{17}{3}$