Question

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

Answer 1

$\frac{1}{2a+b+2x}=\frac{1}{2a}+\frac{1}{b}+\frac{1}{2x}$

$\frac{1}{2a+b+2x}+\frac{1}{2x}=\frac{2a+b}{2ab}$

$\frac{2x+2a+b+2x}{2x.(2a+b+2x)}=\frac{2a+b}{2ab}$

$\frac{2a+b+4x}{2ax+bx+2x^2}=\frac{2a+b}{ab}$

$2a^{2}b+a{b}^2+4abx=4a^2+2abx+4a{x}^2+2abx+b^{2}x+2b{x}^2$

$2a^{2}b+a{b}^2=x^2(4a+2b)+x(4a^2+b^2)$

$x=\pm \left[\frac{-(4a^2+b^2)+\sqrt{(4a^2+b^2{)}^2}+4(4a+2b)(2a^{2}b+a{b}^2)}{2(4a+2b)}\right]$