Find the roots of the following quadratic equation:
$x^{2} -3\sqrt {5}\ x+10=0$

Given: The expression:  $x^{2} -3\sqrt{5} \ x+10=0$

To do: To find the roots of the given quadratic equation.

Solution: We know that for a quadratic equation $ax^{2} +bx+c=0$

 $x=\frac{-b\pm \sqrt{b^{2} -4ac}}{2a}$

On comparing it to the given quadratic equation $a=1,b=-3\sqrt{5} \ and\ c=10$

On substituting these values of $\displaystyle a,\ b\ and\ c$

$x=\frac{-( -3\surd 5) \pm \sqrt{( -3\surd 5)^{2} -4\times 1\times 10}}{2\times 1}$

$x=\frac{3\sqrt{5} \pm \sqrt{( 45-40)}}{2}$

$x=\frac{\left( 3\sqrt{5} \pm \sqrt{5}\right)}{2}$

If $x=\frac{\left( 3\sqrt{5} +\sqrt{5}\right)}{2}$

$\Rightarrow x=\frac{4\sqrt{5}}{2} $

$\Rightarrow x=2\sqrt{5}$

If $x=\frac{\left( 3\sqrt{5} -\sqrt{5}\right)}{2}$

$\Rightarrow x=\frac{\left( 2\sqrt{5}\right)}{2}$

$\Rightarrow x=\sqrt{5}$

$\therefore x=2\sqrt{5}, \ \sqrt{5}$