Question

Solve each of the following equation by using the method of completing the square:
$4x^2+4bx-(a^2-b^2)=0$

Answer 1

Dividing the entire equation by 4

$x^2+bx-\frac{(a²-b²)}{4}=0$

Adding and Subtracting the equation by ${\frac{b}{4}}^2$

$x²+bx+{\left(\frac{b}{2}\right)}^2={\left(\frac{b}{2}\right)}^2+\frac{(a²-b²)}{4}$

$x²+bx+{\left(\frac{b}{2}\right)}^2$this quadratic equation can be written as perfect square as ${\left(\frac{x+b}{2}\right)}^2$

${\left(\frac{x+b}{2}\right)}^2={\frac{b}{4}}^2+{\frac{a}{4}}^2-\frac{b²}{4}$

${\left(\frac{x+b}{2}\right)}^2=\frac{a²}{4}$

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

$x=\frac{(a-b)}{2}$

$x=\frac{(-a-b)}{2}$

The roots are$\frac{a-b}{2}$ and $\frac{-a-b}{2}$