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Solve the following for x; $\frac{1}{2a+b+2x} =\frac{1}{2a} +\frac{1}{b} +\frac{1}{2x}$.
Given: Expression,
$\frac{1}{2a+b+2x} =\frac{1}{2a} +\frac{1}{b} +\frac{1}{2x}$
To do: To solve the equation for x.
Solution: given expression,
$\frac{1}{2a+b+2x} =\frac{1}{2a} +\frac{1}{b} +\frac{1}{2x}$
$\Rightarrow \frac{1}{2a+b+2x} -\frac{1}{2x} =\frac{1}{2a} +\frac{1}{b}$
$\Rightarrow \frac{2x-2a-b-2x}{2x( 2a+b+2x)} =\frac{b+2a}{2ab}$
$\Rightarrow \frac{-( 2a+b)}{2x( 2a+b+2x)} =\frac{2a+b}{2ab}$
$\Rightarrow \frac{-1}{2x( 2a+b+2x)} =\frac{1}{2ab}$
$\Rightarrow 2x( 2a+b+2x) =-2ab$
$\Rightarrow x( 2a+b+2x) =-ab$
$\Rightarrow 2x^{2} +bx+2ax+ab=0$
$\Rightarrow 2x^{2} +2ax+bx+ab=0$
$\Rightarrow 2x( x+a) +b( x+a) =0$
$\Rightarrow ( 2x+b)( x+a) =0$
if $2x+b=0$
$\Rightarrow 2x=-b$
$\Rightarrow x=-\frac{b}{2}$
If $x+a=0$
$\Rightarrow x=-a$
Therefore,The given expression has two solutions for x.
$x=-a$ or $x=-\frac{b}{2}$.
Updated on: 10-Oct-2022
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