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

$2 x^{2} + a x - a^{2} = 0$ [CBSE 2015]

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Solution

The given equation is $2 x^{2} + a x - a^{2} = 0$ .

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

A = 2, B = a and C = $- a^{2}$

∴ Discriminant, D = $B^{2} - 4 A C = a^{2} - 4 \times 2 \times - a^{2} = a^{2} + 8 a^{2} = 9 a^{2} \geq 0$

So, the given equation has real roots.

Now, $\sqrt{D} = \sqrt{9 a^{2}} = 3 a$

$∴ α = \frac{- B + \sqrt{D}}{2 A} = \frac{- a + 3 a}{2 \times 2} = \frac{2 a}{4} = \frac{a}{2} β = \frac{- B - \sqrt{D}}{2 A} = \frac{- a - 3 a}{2 \times 2} = \frac{- 4 a}{4} = - a$

Hence, $\frac{a}{2}$ and $- a$ are the roots of the given equation.

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