Question

Find the roots of the quadratic equation $\sqrt{2}{x}^2+7x+5\sqrt{2}=0$.

Answer 1

We write, $7x=5x+2x\mathrm{as}$ $\sqrt{2}{x}^2\times 5\sqrt{2}=10x^2=5x\times 2x$

$$\begin{gathered} \therefore \sqrt{2}{x}^2+7x+5\sqrt{2}=0 \\ \Rightarrow \sqrt{2}{x}^2+5x+2x+5\sqrt{2}=0 \\ \Rightarrow x\left(\sqrt{2}x+5\right)+\sqrt{2}\left(\sqrt{2}x+5\right)=0 \\ \Rightarrow \left(\sqrt{2}x+5\right)\left(x+\sqrt{2}\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x+\sqrt{2}=0\mathrm{or}\sqrt{2}x+5=0 \\ \Rightarrow x=-\sqrt{2}\mathrm{or}x=-\frac{5}{\sqrt{2}}=-\frac{5\sqrt{2}}{2} \end{gathered}$$
Hence, the roots of the given equation are $-\sqrt{2}\mathrm{and}$ $-\frac{5\sqrt{2}}{2}$.