Question

Find the roots of the given equation $4x^2+4bx-(a^2-b^2)=0$ by the method of completing the square.

Answer 1

We have
$4x^2+4bx-(a^2-b^2)=0$

$\Rightarrow x^2+bx-\frac{a^2-b^2}{4}=0$

$\Rightarrow x^2+2\left(\frac{b}{2}\right)x+{\left(\frac{b}{2}\right)}^2=\frac{a^2-b^2}{4}+{\left(\frac{b}{2}\right)}^2\Rightarrow {(x+\frac{b}{2})}^2=\frac{a^2}{4}$

$\Rightarrow x+\frac{b}{2}=\pm \frac{a}{2}\Rightarrow x=\frac{-b}{2}\pm \frac{a}{2}\Rightarrow x=\frac{-b-a}{2},\frac{-b+a}{2}$

Hence, the roots are $-\left(\frac{a+b}{2}\right)$ and $\left(\frac{a-b}{2}\right)$