Find whether the following equations have real roots. If real roots exist, find them.
$ x^{2}+5 \sqrt{5} x-70=0 $

Given:

Given quadratic equation is \( x^{2}+5 \sqrt{5} x-70=0 \).

To do:

We have to determine whether the given quadratic equation has real roots.

Solution:

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=1, b=5 \sqrt{5}$ and $c=-70$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is

$D=b^2-4ac$.

Therefore,

$D=(5 \sqrt{5})^2-4(1)(-70)$

$=125+280$

$=405$.

As $D>0$, the given quadratic equation has two distinct real roots.

This implies,

$x=\frac{-b\pm \sqrt{D}}{2a}$

$x=\frac{-5 \sqrt{5} \pm \sqrt{405}}{2(1)}$ 

$x=\frac{-5 \sqrt{5}\pm 9\sqrt5}{2}$ 

$x=\frac{-5 \sqrt{5}+9 \sqrt{5}}{2}$ or $x= \frac{-5 \sqrt{5}-9 \sqrt{5}}{2}$

$x=\frac{4\sqrt{5}}{2}$ or $x=\frac{-14 \sqrt{5}}{2}$

$x=2\sqrt5$ or $x=-7\sqrt5$

The roots of the given quadratic equation are $-7\sqrt5$ and $2\sqrt5$.