Find whether the following equations have real roots. If real roots exist, find them.
$ -2 x^{2}+3 x+2=0 $

Given:

Given quadratic equation is \( -2 x^{2}+3 x+2=0 \)

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,

$a=-2, b=3$ and $c=2$.

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

$D=b^2-4ac$.

Therefore,

$D=(3)^2-4(-2)(2)$

$=9+16$

$=25$.

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

This implies,

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

$x=\frac{-3 \pm \sqrt{25}}{2(-2)}$ 

$x=\frac{-3 \pm 5}{-4}$ 

$x=\frac{-3+5}{-4}$ or $x= \frac{-3-5}{-4}$

$x=\frac{2}{-4}$ or $x=\frac{-8}{-4}$

$x=-\frac{1}{2}$ or $x=2$

The roots of the given quadratic equation are $-\frac{1}{2}$ and $2$. 

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Updated on: 10-Oct-2022

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