Solve the following quadratic equations by factorization:

We have been given

$a^{2} x^{2}-3 a b x+2 b^{2}=0$

$a^{2} x^{2}-2 a b x-a b x+2 b^{2}=0$

$a x(a x-2 b)-b(a x-2 b)=0$

$(a x-b)(a x-2 b)=0$

Therefore,

$a x-b=0$

$a x=b$

$x=\frac{b}{a}$

or,

$a x-2 b=0$

$a x=2 b$

$x=\frac{2 b}{a}$

Hence, $x=\frac{b}{a}$ or $x=\frac{2 b}{a}$