Question

Find the roots of the following equations:
(i) $\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30},x\ne -4,7$

Answer 1

$\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}$

$\Rightarrow \frac{x-7-x-4}{x^2-3x-28}=\frac{11}{30}$

$\Rightarrow \frac{-11}{x^2-3x-28}=\frac{11}{30}$

$\Rightarrow \frac{-1}{x^2-3x-28}=\frac{1}{30}$

$\Rightarrow x^2-3x-28=-30$

$\Rightarrow x^2-3x-28+30=0$

$\Rightarrow x^2-3x+2=0$

$\Rightarrow x^2-2x-x+2=0$

$\Rightarrow x(x-2)-1(x-2)=0$

$\Rightarrow (x-2)(x-1)=0$

$\therefore x=1,2$

Roots are $\{1,2\}$