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

$x^{2} - 2 a x - (4 b^{2} - a^{2}) = 0$ [CBSE 2015]

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Solution

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

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

A = 1, B = $- 2 a$ and C = $- (4 b^{2} - a^{2})$

∴ Discriminant, D = $B^{2} - 4 A C = {(- 2 a)}^{2} - 4 \times 1 \times [- (4 b^{2} - a^{2})] = 4 a^{2} + 16 b^{2} - 4 a^{2} = 16 b^{2} > 0$

So, the given equation has real roots.

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

$∴ α = \frac{- B + \sqrt{D}}{2 A} = \frac{- (- 2 a) + 4 b}{2 \times 1} = \frac{2 (a + 2 b)}{2} = a + 2 b β = \frac{- B - \sqrt{D}}{2 A} = \frac{- (- 2 a) - 4 b}{2 \times 1} = \frac{2 (a - 2 b)}{2} = a - 2 b$

Hence, $a + 2 b$ and $a - 2 b$ are the roots of the given equation.

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