Question

Find the roots of the quadratic equation using completing the square method: $4x^2-4px+p^2-q^2=0$

Answer 1

$4x^2-4px+p^2-q^2=0$

$\Rightarrow$ $x^2-px+\frac{p^2}{4}-\frac{q^2}{4}=0$ [ Dividing both sides by $4$ ]

Here, $a=1,b=-p,c=\frac{p^2}{4}-\frac{q^2}{4}$

$\Rightarrow$ $x^2-px=-\frac{p^2}{4}+\frac{q^2}{4}$

Now, ${\left(\frac{b}{2}\right)}^2={\left(\frac{-p}{2}\right)}^2=\frac{p^2}{4}$

Now, we are adding $\frac{p^2}{4}$ on both side of equation.

$\Rightarrow$ $x^2-px+\frac{p^2}{4}=-\frac{p^2}{4}+\frac{q^2}{4}+\frac{p^2}{4}$

$\Rightarrow$ ${(x-\frac{p}{2})}^2=\frac{q^2}{4}$

$\Rightarrow$ $x-\frac{p}{2}=\pm \frac{q}{2}$

$\therefore$ $x=\frac{q+p}{2}$ and $x=\frac{-q+p}{2}$