Q. If the AM and GM of roots of a quadratic equations are 8 and 5 respectively, then the quadratic equation will be
A) $x^{2} - 16x - 25 = 0$
B) $x^{2} - 8x + 5 = 0$
C) $x^{2} - 16x + 25 = 0$
D) $x^{2} + 16x - 25 = 0$
Explanation
(c)
Given that, AM =8, GM =5, if $\alpha,\ \beta$ are the roots of quadratic equation, then the required quadratic equation is
$x^{2} - x(\alpha + \beta) + \alpha\beta = 0$ …(i)
Here, AM $= \frac{\alpha + \beta}{2} = 8$ $\Rightarrow \ \alpha + \beta = 16$
And GM =$\ \sqrt{\alpha\beta} = 5$ $\Rightarrow \ \alpha\beta = 25$
From Eq. ( I )
$x^{2} - 16x + 25 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.