# If A and G are the arithmetic and geometric means, respectively,

Question:

If A and G are the arithmetic and geometric means, respectively, of the roots of a quadratic equation. Then, the equation is _________.

Solution:

Let A represent arithmetic mean and G represent geometric mean of the roots of a quadratic equation.

Let us suppose the roots of the equation be a and b

According t o given condition:-

$A \cdot M=\frac{a+b}{2}=A$ i. e $a+b=2 A$

$G . M=\sqrt{a b}=G$ i. e $a b=G^{2}$

Hence, quadratic equation is given by x– (a + b)x + ab = 0

[x– (sum of roots) x + (product of roots = 0)

i.e x– 2Ax + G2 = 0