Find x, if 16 (a-x a+x )^3=a+x a-x - Vidyadip

Question

Find x, if $16\left(\frac{a-x}{a+x}\right)^3=\frac{a+x}{a-x}$

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Answer

$\left(\frac{a-x}{a+x}\right)^3=\frac{a+x}{a-x}$
$\Rightarrow 16=\left(\frac{a-x}{a+x}\right)^4 $
$ \Rightarrow(2)^4=\left(\frac{a-x}{a+x}\right)^4$
$\Rightarrow \frac{a-x}{a+x}= \pm 2$
$ \Rightarrow \frac{a-x}{a+x}=\frac{2}{1} \text { or } \frac{a-x}{a+x}=\frac{-2}{1}$
Applying componendo and Dividendo, we get
$\Rightarrow \frac{ a + x + a - x }{ a + x - a + x }=\frac{3}{1} \text { or } \frac{ a + x + a - x }{ a + x - a + x }=\frac{-1}{-3} $
$ \Rightarrow \frac{2 a }{2 x }=3 \text { or } \frac{2 a }{2 x }=\frac{1}{3}$
$ \Rightarrow x =\frac{ a }{3} \text { or } x =3 a $

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