# The value of c for which the equation ax2 + 2bx + c = 0 has equal roots is

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)