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Find the roots of the quadratic equations by using the quadratic formula in each of the following:
$ -x^{2}+7 x-10=0 $
Given:
Given quadratic equation is \( -x^{2}+7 x-10=0 \).
To do:
We have to find the roots of the given quadratic equation.
Solution:
$-x^2+7x - 10 = 0$
The above equation is of the form $ax^2 + bx + c = 0$, where $a = -1, b = 7$ and $c =-10$
Discriminant $\mathrm{D} =b^{2}-4 a c$
$=(7)^{2}-4 \times (-1)\times(-10)$
$=49-40$
$=9$
$\mathrm{D}>0$
Let the roots of the equation are $\alpha$ and $\beta$
$\alpha =\frac{-b+\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-7+\sqrt{9}}{2(-1)}$
$=\frac{-7+3}{-2}$
$=\frac{-4}{-2}$
$=2$
$\beta =\frac{-b-\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-7-\sqrt{9}}{2(-1)}$
$=\frac{-7-3}{-2}$
$=\frac{-10}{-2}$
$=5$
Hence, the roots of the given quadratic equation are $2, 5$.
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