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

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 $C=-a^{2}$

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

$\therefore \alpha=\frac{-B+\sqrt{D}}{2 A}=\frac{-a+3 a}{2 \times 2}=\frac{2 a}{4}=\frac{a}{2}$

$\beta=\frac{-B-\sqrt{D}}{2 A}=\frac{-a-3 a}{2 \times 2}=\frac{-4 a}{4}=-a$

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