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

Given: The quadratic equation $2x^{2} +ax-a^{2} =0$.

To do: To solve the the given quadratic equation for $x$.

As given the quadratic equation $2x^{2} +ax-a^{2} =0$ ,

on comparing the given quadratic equation to $ax^{2} +bx+c=0$,

We have ${a} =2,\ b=a$ and $c=-a^{2}$

We know for a quadratic equation $x=\frac{-b\pm \sqrt{b^{2} -4ac}}{2a}$

By using the above quadratic formula,

$x=\frac{-a\pm \sqrt{a^{2} -4\times 2\times ( -a)^{2}}}{2\times 2}$

$\Rightarrow x=\frac{-a\pm \sqrt{a^{2} +8a^{2}}}{4}$

$\Rightarrow x=\frac{-a\pm \sqrt{9a^{2}}}{4}$

$\Rightarrow x=\frac{-a\pm 3a}{4}$

$\Rightarrow x=\frac{-a+3a}{4} \ or\ \frac{-a-3a}{4}$

$\Rightarrow x=\frac{2a}{4} \ or\ \frac{-4a}{4}$

$\Rightarrow x=\frac{a}{2}$ or $-a$

Therefore, The solutions for the given quadratic equation are $x=\frac{a}{2}$ or $-a$.