Question

Solve the equation $a^2{x}^2-3abx+2b^2=0$ by the method of completing the square.

Answer 1

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

$(ax{)}^2-2(ax)\left(\frac{3b}{2}\right)+2b^2+{\left(\frac{3b}{2}\right)}^2={\left(\frac{3b}{2}\right)}^2$

$(ax{)}^2-2(ax)\left(\frac{3b}{2}\right)+{\left(\frac{3b}{2}\right)}^2=\frac{9b^2}{4}-2b^2$

${[ax-\frac{3b}{2}]}^2=\frac{9b^2-8b^2}{4}$

$(ax-\frac{3b}{2})=\pm \sqrt{\frac{b^2}{4}}=\pm \frac{b}{2}$

$ax=\frac{3b}{2}+\frac{b}{2}$; $ax=\frac{3b}{2}-\frac{b}{2}$

$ax=\frac{4b}{2}$; $ax=\frac{2b}{2}$

$ax=2b$; $ax=b$

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