Question

Solve each of the following quadratic equations:
$\frac{a}{(ax-1)}+\frac{b}{(bx-1)}=(a+b),x\ne \frac{1}{a},\frac{1}{b}$

Answer 1

$\frac{a}{(ax-1)}+\frac{b}{(bx-1)}=(a+b)$

$\Rightarrow \frac{a}{(ax-1)}-b=a-\frac{b}{(bx-1)}$

$\Rightarrow \frac{a-abx+b}{(ax-1)}=\frac{abx-a-b}{(bx-1)}$

$\Rightarrow (a+b-abx)(bx-1)=-(a+b-abx)(ax-1)$

$\Rightarrow (a+b-abx)(bx-1)+(a+b-abx)(ax-1)=0$

$\Rightarrow (a+b-abx)(bx-1+ax-1)=0$

$\Rightarrow (a+b-abx)(ax+bx-2)=0$

$\Rightarrow a+b-abx=0$ or $ax+bx-2=0$

$\Rightarrow abx=a+b$ or $(a+b)x=2$

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