Question:
The value of $c$ for which the equation $a x^{2}+2 b x+c=0$ has equal roots is
(a) $\frac{b^{2}}{a}$
(b) $\frac{b^{2}}{4 a}$
(c) $\frac{a^{2}}{b}$
(d) $\frac{a^{2}}{4 b}$
Solution:
The given quadric equation is $a x^{2}+2 b x+c=0$, 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
$\alpha+\beta=\frac{-2 b}{a}$
$\alpha+\alpha=\frac{-2 b}{a}$
$2 a=\frac{-2 b}{a}$
$\alpha=\frac{-b}{a}$
And the product of the roots
$\alpha \cdot \beta=\frac{c}{a}$
$\alpha \alpha=\frac{c}{a}$
Putting the value of $\alpha$
$\frac{-b}{a} \times \frac{-b}{a}=\frac{c}{a}$
$\frac{b^{2}}{a}=c$
Therefore, the value of $c=\frac{b^{2}}{a}$
Thus, the correct answer is (a)