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 the root of the quadratic equation be $a$ and $b$ .
According to given condition ,
$\text{A.M.}=\frac{a+b}{2}=8\Rightarrow a+b=16...\left(1\right)\text{G.M.}=\sqrt{ab}=5\Rightarrow ab=25...\left(2\right)$
Thus the quadratic equation is given by,
$x^2-x\left(\text{sum of roots}\right)+\left(\text{product of roots}\right)=0x^2-x(a+b)+\left(ab\right)=0\Rightarrow x^2-16x+25=0$