Question

Solve equations, using factorization method :
$\frac{4}{x+2}-\frac{1}{x+3}=\frac{4}{2x+1}$

Answer 1

Step 1: Simplifying the given equation:
Simplify the given equation
Given,
$\frac{4}{x+2}-\frac{1}{x+3}=\frac{4}{2x+1}$
$\Rightarrow \frac{4(x+3)-1(x+2)}{(x+2)(x+3)}=\frac{4}{2x+1}$
$\Rightarrow \frac{4x+12-x-2}{x^2+2x+3x+6}=\frac{4}{2x+1}$
$\Rightarrow \frac{3x+10}{x^2+5x+6}=\frac{4}{2x+1}$
$\Rightarrow (3x+10)(2x+1)=4(x^2+5x+6)$
$\Rightarrow 6x^2+3x+20x+10=4x^2+20x+24$
$\Rightarrow 2x^2+3x-14=0$

Step 2 : Factorizing the above equation :
Factorize the above equation
$\Rightarrow 2x^2+7x-4x-14=0\left\{\text{factorizing left hand side}\right\}$
$\Rightarrow x(2x+7)-2(2x+7)=0$
$\Rightarrow (2x+7)(x-2)=0$
$\Rightarrow 2x+7=0$ or $x-2=0\left\{\text{zero product rule}\right\}$
$\therefore x=-\frac{7}{2}$ or $x=2$
Hence, the required answer is $x=\frac{7}{2}$ or $x=2$.