Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: $2 x^{2}+a x-a^{2}=0$.

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Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: $2 x^{2}+a x-a^{2}=0$ .

Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: $2 x^{2}+a x-a^{2}=0$ .

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, $\sqrt{D}=\sqrt{9 a^{2}}=3 a$
$\begin{array}{l} \therefore \alpha=\frac{-B+\sqrt{D}}{2 A}=\frac{-a+3 a}{2 \times 2}=\frac{2 a}{4}=\frac{a}{2} \\ \beta=\frac{-B-\sqrt{D}}{2 A}=\frac{-a-3 a}{2 \times 2}=\frac{-4 a}{4}=-a \end{array}$