Solve the quadratic equation $2 x^{2} + a x - a^{2} = 0$ for x.

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Solution

We have : $2 x^{2} + a x - a^{2} = 0$
Comparing the given equation with the standard quadratic equation
$(a x^{2} + b x + c = 0)$ , we get
$a = 2, b = a, c = - a^{2}$
Using the quadratic formula, $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , we get :
$x = \frac{- a \pm \sqrt{a^{2} - 4 \times 2 \times (- a)^{2}}}{2 \times 2}$
$= \frac{- a \pm \sqrt{9 a^{2}}}{4}$
$= \frac{- a \pm 3 a}{4}$
$\Rightarrow x = \frac{- a + 3 a}{4} = \frac{a}{2}$ or $x = \frac{a - 3 a}{4} = - a$
So, the solutions of the given quadratic equation are $x = \frac{a}{2}$ or $x = - a .$

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