Question

Solve each of the following quadratic equations:

$2x^2+ax-a^2=0$ [CBSE 2015]

Answer 1

We write, $ax=2ax-ax$ as $2x^2\times \left(-a^2\right)=-2a^2{x}^2=2ax\times \left(-ax\right)$
$$\begin{gathered} \therefore 2x^2+ax-a^2=0 \\ \Rightarrow 2x^2+2ax-ax-a^2=0 \\ \Rightarrow 2x\left(x+a\right)-a\left(x+a\right)=0 \\ \Rightarrow \left(x+a\right)\left(2x-a\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x+a=0\mathrm{or}2x-a=0 \\ \Rightarrow x=-a\mathrm{or}x=\frac{a}{2} \end{gathered}$$
Hence, $-a$ and $\frac{a}{2}$ are the roots of the given equation.