Question

What should be added to $\frac{a}{a-b}+\frac{b}{a+b}$ to get $1?$

Answer 1

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

find k

$\Rightarrow \frac{a(a+b)+b(a-b)}{(a+b)(a-b)}+k=1$

$\Rightarrow \frac{a^2+ab+ab-b^2}{a^2-b^2}+k=1$

$\Rightarrow k=1-\frac{(a^2-b^2+2ab)}{a^2-b^2}=\frac{a^2-b^2-a^2+b^2-2ab}{a^2-b^2}$

$\Rightarrow k=\frac{2ab}{b^2-a^2}$

$\therefore$ we need to add $\frac{2ab}{b^2-a^2}$