Question

Solve for $x$:
$\frac{1}{x+1}+\frac{2}{x+2}=\frac{1}{x+4},x\ne -1,-2,-4$.

Answer 1

$\frac{1}{x+1}+\frac{2}{x+2}=\frac{1}{x+4}$

$\Rightarrow \frac{x+2+2x+2}{(x+1)(x+2)}=\frac{1}{x+4}$

$\Rightarrow (3x+4)(x+4)=(x+1)(x+2)$

$\Rightarrow 3x^2+12x+4x+16=x^2+2x+x+2$

$\Rightarrow 2x^2+13x+14=0$

$x=\frac{-13\pm \sqrt{{13}^2-4\times 2\times 14}}{2\times 2}$

$\therefore x=\frac{-13\pm \sqrt{57}}{4}$