Question

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

$4x^2+4bx-\left(a^2-b^2\right)=0$ [CBSE 2015]

Answer 1

The given equation is $4x^2+4bx-\left(a^2-b^2\right)=0$.

Comparing it with $A{x}^2+Bx+C=0$, we get

A = 4, B = 4b and C = $-\left(a^2-b^2\right)$

∴ Discriminant, D = $B^2-4AC={\left(4b\right)}^2-4\times 4\times \left[-\left(a^2-b^2\right)\right]=16b^2+16a^2-16b^2=16a^2>0$

So, the given equation has real roots.

Now, $\sqrt{D}=\sqrt{16a^2}=4a$

$$\begin{gathered} \therefore \alpha =\frac{-B+\sqrt{D}}{2A}=\frac{-4b+4a}{2\times 4}=\frac{4\left(a-b\right)}{8}=\frac{a-b}{2} \\ \beta =\frac{-B-\sqrt{D}}{2A}=\frac{-4b-4a}{2\times 4}=\frac{-4\left(a+b\right)}{8}=-\frac{a+b}{2} \end{gathered}$$

Hence, $\frac{1}{2}\left(a-b\right)$ and $-\frac{1}{2}\left(a+b\right)$ are the roots of the given equation.