Solve the following for x : 1 2 a+b+2 x =1 2 a +1 b +1 2 x - Vidyadip

Question

Solve the following for $x$ :
$\frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x}$

✓

Answer

$\begin{array}{l}\frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x} \\ \frac{1}{2 a+b+2 x}-\frac{1}{2 x}=\frac{1}{2 a}+\frac{1}{b} \\ \Rightarrow \frac{2 x-2 a-b-2 x}{(2 x)(2 a+b+2 x)}=\frac{(2 a+b)}{2 a b} \\ \Rightarrow \frac{-(2 a+b)}{(2 x)(2 a+b+2 x)}=\frac{(2 a+b)}{2 a b} \\ \Rightarrow-2 a b=(2 x)(2 a+b+2 x) \\ \Rightarrow-2 a b=4 a x+2 b x+4 x^2 \\ \Rightarrow 4 x^2+2 b x+4 a x+2 a b=0 \\ \Rightarrow 2 x(2 x+b)+2 a(2 x+b)=0\end{array}$
$\begin{array}{l}\Rightarrow(2 x+2 a)(2 x+b)=0 \\2 x+2 a=0 \Rightarrow x=-a\end{array}$
And $2 x+b=0 \Rightarrow x=\frac{-b}{2}$
Hence, $\quad x=-a, \frac{-b}{2}$

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