Question

Solve by factorization:
$\frac{1}{a+b+x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{x},a+b\ne 0$

Answer 1

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

$\Rightarrow \frac{-(a+b)}{x(a+b+x)}=\frac{(a+b)}{ab}$

$\Rightarrow -ab=x(a+b+x)$

$\Rightarrow x^2+(a+b)x+ab=0$

$\Rightarrow x^2+ax+bx+ab=0$

$\Rightarrow x(x+a)+b(x+a)=0$

$\Rightarrow (x+b)(x+a)=0$

$\Rightarrow x=-a$ or $x=-b$.