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

Question:

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

$4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0$

 

Solution:

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

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

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

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

So, the given equation has real roots.

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

$\therefore \alpha=\frac{-B+\sqrt{D}}{2 A}=\frac{-4 b+4 a}{2 \times 4}=\frac{4(a-b)}{8}=\frac{a-b}{2}$

$\beta=\frac{-B-\sqrt{D}}{2 A}=\frac{-4 b-4 a}{2 \times 4}=\frac{-4(a+b)}{8}=-\frac{a+b}{2}$

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

 

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