If ax^2 + bx + c = 0 has equal roots, then c = - Vidyadip

MCQ

If $ax^2 + bx + c = 0$ has equal roots, then $c =$

A
$\frac{-\text{b}}{2\text{a}}$
B
$\frac{\text{b}}{2\text{a}}$
C
$\frac{-\text{b}^2}{4\text{a}}$
✓
$\frac{-\text{b}^2}{4\text{a}}$

✓

Answer

Correct option: D.

$\frac{-\text{b}^2}{4\text{a}}$

The given quadric equation is $ax^2 + bx + c = 0$, 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
$\alpha+\beta=\frac{-\text{b}}{\text{a}}$
$\alpha+\alpha=\frac{-\text{b}}{\text{a}}$
$2\alpha=\frac{-\text{b}}{\text{a}}$
$\alpha=\frac{-\text{b}}{2\text{a}}$
And the product of the roots
$\alpha\cdot\beta=\frac{\text{c}}{\text{a}}$
$\alpha\alpha=\frac{\text{c}}{\text{a}}$
Putting the value of $\alpha$
$\frac{-\text{b}}{2\text{a}}\times\frac{-\text{b}}{2\text{a}}=\frac{\text{c}}{\text{a}}$
$\frac{\text{b}^2}{4\text{a}}=\text{c}$
Therefore, the value of $\text{c}=\frac{\text{b}^2}{4\text{a}}$
Thus, the correct answer is $(d)$

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