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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