Question

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

Answer 1

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:-
$$\begin{gathered} A.M=\frac{a+b}{2}=A\mathrm{i}.\mathrm{e}a+b=2A \\ G.M=\sqrt{ab}=G\mathrm{i}.\mathrm{e}ab=G^2 \end{gathered}$$
Hence, quadratic equation is given by x² – (a + b)x + ab = 0
[x² – (sum of roots) x + (product of roots = 0)
i.e x² – 2Ax + G² = 0