Question:
If $a x^{2}+b x+c=0$ has equal roots, then $c=$
(a) $\frac{-b}{2 a}$
(b) $\frac{b}{2 a}$
(c) $\frac{-b^{2}}{4 a}$
(d) $\frac{b^{2}}{4 a}$
Solution:
The given quadric equation is $a x^{2}+b x+c=0$, and roots are equal
Then find the value of c.
Let $\alpha$ and $\beta$ be two roots of given equation $\alpha=\beta$
Then, as we know that sum of the roots
$\alpha+\beta=\frac{-b}{a}$
$\alpha+\alpha=\frac{-b}{a}$
$2 \alpha=\frac{-b}{a}$
$a=\frac{-b}{2 a}$
And the product of the roots
$\alpha \cdot \beta=\frac{c}{a}$
$\alpha \alpha=\frac{c}{a}$
Putting the value of $\alpha$
$\frac{-b}{2 a} \times \frac{-b}{2 a}=\frac{c}{a}$
$\frac{b^{2}}{4 a}=c$
Therefore, the value of $c=\frac{b^{2}}{4 a}$
Thus, the correct answer is (d)