Q. If $x^{2} + px + q = 0$ has the roots $\alpha\ and\ \beta$, then the value of $(\alpha - \beta)^{2}$ is equal to
A) $p^{2} - 4q$
B) $\left( p^{2} - 4q \right)^{2}$
C) $p^{2} + 4q$
D) $\left( p^{2} + 4q \right)^{2}$
Explanation
(a)
$\alpha + \beta = - p,\ \alpha\beta = q$
$\therefore\ \ \ \alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta$
$= p^{2} - 2q$
$\Rightarrow (\alpha - \beta)^{2} = \alpha^{2} + \beta^{2} - 2\alpha\beta$
$= \left( p^{2} - 2q \right) + 2q$
$= p^{2} - 4q$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.