Question

If $A.M.$ and $G.M.$ of roots of a quadratic equation are $8$ and $5$. respectively, then obtain the quadratic equation.

Answer 1

Let $\alpha \&\beta$ be the roots of the quadratic equation

It is given that

$A.M.=\frac{\alpha +\beta }{2}=8$ and $\sqrt{\alpha \beta }=5$

Thus, $\alpha +\beta =16$ $...\left(1\right)$

$\alpha \beta =25$ $...\left(2\right)$

Now, quadratic equation is

$x^2$ -Sum of roots)$x$ + [Product of roots] = $0$

$x^2-(\alpha +\beta )x+\left(\alpha \beta \right)=0$

Substituing $\alpha +\beta =16,\alpha \beta =25$

from $\left(1\right)$ & $\left(2\right)$

$x^2-16x+25=0$

Thus, the required quadratic equation is

$x^2-16x+25=0$