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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