# Solve each of the following quadratic equations

Question:

Solve each of the following quadratic equations

$3\left(\frac{3 x-1}{2 x+3}\right)-2\left(\frac{2 x+3}{3 x-1}\right)=5, \quad x \neq \frac{1}{3},-\frac{3}{2}$

Solution:

$3\left(\frac{3 x-1}{2 x+3}\right)-2\left(\frac{2 x+3}{3 x-1}\right)=5, \quad x \neq \frac{1}{3},-\frac{3}{2}$

$\Rightarrow \frac{3(3 x-1)^{2}-2(2 x+3)^{2}}{(2 x+3)(3 x-1)}=5$

$\Rightarrow \frac{3\left(9 x^{2}-6 x+1\right)-2\left(4 x^{2}+12 x+9\right)}{6 x^{2}+7 x-3}=5$

$\Rightarrow \frac{27 x^{2}-18 x+3-8 x^{2}-24 x-18}{6 x^{2}+7 x-3}=5$

$\Rightarrow \frac{19 x^{2}-42 x-15}{6 x^{2}+7 x-3}=5$

$\Rightarrow 19 x^{2}-42 x-15=30 x^{2}+35 x-15$

$\Rightarrow 11 x^{2}+77 x=0$

$\Rightarrow 11 x(x+7)=0$

$\Rightarrow x=0$ or $x+7=0$

$\Rightarrow x=0$ or $x=-7$

Hence, 0 and −7 are the roots of the given equation.