If A.M. and G.M. of roots of a quadratic equation are 8 and 5,

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

Solution:

Let the root of the quadratic equation be a and b.

According to the given condition,

A.M. $=\frac{a+b}{2}=8 \Rightarrow a+b=16$ $\ldots(1)$

G.M. $=\sqrt{a b}=5 \Rightarrow a b=25$ $\ldots(2)$

G.M. $=\sqrt{a b}=5 \Rightarrow a b=25$

The quadratic equation is given by,

$x^{2}-x$ (Sum of roots) $+$ (Product of roots) $=0$

$x^{2}-x(a+b)+(a b)=0$

$x^{2}-16 x+25=0[U \operatorname{sing}(1)$ and $(2)]$

Thus, the required quadratic equation is $x^{2}-16 x+25=0$