Question

Solve equations, using factorization method:
$\frac{5}{x-2}-\frac{3}{x+6}=\frac{4}{x}$

Answer 1

Step 1: Simplifying the given equation :
Simplify the given equation
Given equation,
$\frac{5}{x-2}-\frac{3}{x+6}=\frac{4}{x}$
$\Rightarrow \frac{5(x+6)-3(x-2)}{(x-2)(x+6)}=\frac{4}{x}$
$\Rightarrow \frac{5x+30-3x+6}{x^2+6x-2x+12}=\frac{4}{x}$
$\Rightarrow \frac{2x+36}{x^2+4x-12}=\frac{4}{x}$
$\Rightarrow x^2+18x=2x^2+8x-24$
$\Rightarrow x^2-10x-24=0$

Step 2: Factorizing the above equation:
Factorize the above equaion
$\Rightarrow x^2-12x+2x-24=0‘\left\{\text{factorizing left hand side}\right\}$
$\Rightarrow x(x-12)+2(x-12)=0$
$\Rightarrow (x-12)(x+2)=0$
$\Rightarrow x-12=0$ or $x+2=0\left\{\text{zero product rule}\right\}$
$\therefore x=12$ or $x=-2$
Hence, the required answer is $x=12$ or $x=-2$.