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Find the roots of the quadratic equations by using the quadratic formula in each of the following:
$ 5 x^{2}+13 x+8=0 $
Given:
Given quadratic equation is \( 5 x^{2}+13 x+8=0 \).
To do:
We have to find the roots of the given quadratic equation.
Solution:
$5x^2+13x + 8 = 0$
The above equation is of the form $ax^2 + bx + c = 0$, where $a = 5, b = 13$ and $c = 8$
Discriminant $\mathrm{D} =b^{2}-4 a c$
$=(13)^{2}-4 \times 5\times8$
$=169-160$
$=9$
$\mathrm{D}>0$
Let the roots of the equation are $\alpha$ and $\beta$
$\alpha =\frac{-b+\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-13+\sqrt{9}}{2(5)}$
$=\frac{-13+3}{10}$
$=\frac{-10}{10}$
$=-1$
$\beta =\frac{-b-\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-13-\sqrt{9}}{2(5)}$
$=\frac{-13-3}{10}$
$=\frac{-16}{10}$
$=-\frac{8}{5}$
Hence, the roots of the given quadratic equation are $-\frac{8}{5}, -1$.
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