# If ax2 + bx + c = 0 has equal roots, then c =

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)