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Find the roots of the quadratic equations by using the quadratic formula in each of the following:
\( -3 x^{2}+5 x+12=0 \)
Given:
Given quadratic equation is \( -3 x^{2}+5 x+12=0 \).
To do:
We have to find the roots of the given quadratic equation.
Solution:
$-3x^2+5x + 12 = 0$
The above equation is of the form $ax^2 + bx + c = 0$, where $a = -3, b = 5$ and $c = 12$
Discriminant $\mathrm{D} =b^{2}-4 a c$
$=(5)^{2}-4 \times (-3)\times12$
$=25+144$
$=169$
$\mathrm{D}>0$
Let the roots of the equation are $\alpha$ and $\beta$
$\alpha =\frac{-b+\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-5+\sqrt{169}}{2(-3)}$
$=\frac{-5+13}{-6}$
$=\frac{8}{-6}$
$=-\frac{4}{3}$
$\beta =\frac{-b-\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-5-\sqrt{169}}{2(-3)}$
$=\frac{-5-13}{-6}$
$=\frac{-18}{-6}$
$=3$
Hence, the roots of the given quadratic equation are $-\frac{4}{3}, 3$.