Question

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

$\sqrt{2}{x}^2+7x+5\sqrt{2}=0$ [CBSE 2013]

Answer 1

The given equation is $\sqrt{2}{x}^2+7x+5\sqrt{2}=0$.

Comparing it with $a{x}^2+bx+c=0$, we get

a = $\sqrt{2}$, b = 7 and c = $5\sqrt{2}$

∴ Discriminant, D = $b^2-4ac={\left(7\right)}^2-4\times \sqrt{2}\times 5\sqrt{2}=49-40=9>0$

So, the given equation has real roots.

Now, $\sqrt{D}=\sqrt{9}=3$

$$\begin{gathered} \therefore \alpha =\frac{-b+\sqrt{D}}{2a}=\frac{-7+3}{2\times \sqrt{2}}=\frac{-4}{2\sqrt{2}}=-\sqrt{2} \\ \beta =\frac{-b-\sqrt{D}}{2a}=\frac{-7-3}{2\times \sqrt{2}}=\frac{-10}{2\sqrt{2}}=-\frac{5\sqrt{2}}{2} \end{gathered}$$

Hence, $-\sqrt{2}$ and $-\frac{5\sqrt{2}}{2}$ are the roots of the given equation.