Question

Write quadratic equation the arithmetic and geometric means of whose roots are A and G respectively.

Answer 1

Let $\alpha ,\beta ,$ be roots of equation, so equation is

$x^2-(\alpha +\beta )x+\alpha \beta =0$

Here,

$\frac{\alpha +\beta }{2}=A$

$\alpha +\beta =2A$

And $\sqrt{\alpha }=G$

$\sqrt{\alpha \beta }=G$

$\alpha \beta =G^2$

So,

Required equation is,

$x^2-2Ax+G^2=0$