Find whether the following equations have real roots. If real roots exist, find them.
$ 5 x^{2}-2 x-10=0 $

Given:

Given quadratic equation is \( 5 x^{2}-2 x-10=0 \)

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,

$a=5, b=-2$ and $c=-10$.

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

$D=b^2-4ac$.

Therefore,

$D=(-2)^2-4(5)(-10)$

$=4+200$

$=204$.

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

This implies,

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

$x=\frac{-(-2) \pm \sqrt{204}}{2(5)}$ 

$x=\frac{2 \pm 2\sqrt{51}}{10}$ 

$x=\frac{2(1+\sqrt{51})}{10}$ or $x= \frac{2(1-\sqrt{51})}{10}$

$x=\frac{1+\sqrt{51}}{5}$ or $x=\frac{1-\sqrt{51}}{5}$

The roots of the given quadratic equation are $\frac{1+\sqrt{51}}{5}$ and $\frac{1-\sqrt{51}}{5}$.  

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Updated on: 10-Oct-2022

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