Question

Solve each of the following quadratic equations:

$x^2+3\sqrt{3}x-30=0$ [CBSE 2015]

Answer 1

We write, $3\sqrt{3}x=5\sqrt{3}x-2\sqrt{3}x$ as $x^2\times \left(-30\right)=-30x^2=5\sqrt{3}x\times \left(-2\sqrt{3}x\right)$

$$\begin{gathered} \therefore x^2+3\sqrt{3}x-30=0 \\ \Rightarrow x^2+5\sqrt{3}x-2\sqrt{3}x-30=0 \\ \Rightarrow x\left(x+5\sqrt{3}\right)-2\sqrt{3}\left(x+5\sqrt{3}\right)=0 \\ \Rightarrow \left(x+5\sqrt{3}\right)\left(x-2\sqrt{3}\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x+5\sqrt{3}=0\mathrm{or}x-2\sqrt{3}=0 \\ \Rightarrow x=-5\sqrt{3}\mathrm{or}x=2\sqrt{3} \end{gathered}$$
Hence, the roots of the given equation are $-5\sqrt{3}$ and $2\sqrt{3}$.