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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 $x$.
Solution:
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 $-a$.
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