If AM and GM of the roots of a quadratic equation are 10 and 8 respectively then obtain the quadratic equation.

To find: The quadratic equation.

Given: (i) AM of roots of quadratic equation is 10

(ii) GM of roots of quadratic equation is 8

Formula used: (i) Arithmetic mean between a and $b=\frac{a+b}{2}$

(ii) Geometric mean between $a$ and $b=\sqrt{a b}$

Let the roots be p and q

Arithmetic mean of roots $p$ and $q=\frac{p+q}{2}=10$

$\Rightarrow \frac{p+q}{2}=10$

⇒ p + q = 20 = sum of roots … (i)

Geometric mean of roots p and q $=\sqrt{p q}=8$

⇒ pq = 64 = product of roots … (ii)

Quadratic equation $=x^{2}-($ sum of roots $) x+($ product of roots $)$

From equation (i) and (ii)

Quadratic equation $=x^{2}-(20) x+(64)$

$=x^{2}-20 x+64$

$x^{2}-20 x+64$