Question

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

Answer 1

Let the roots of the required quadratic equation be a and b. 

The arithmetic and geometric means of roots are A and G respectively.

⇒ A = (a + b)/2…(i)

And G = \(\sqrt{ab}\)…(ii)

We know that the equation whose roots are given is =

x²- (a+b)x + ab = 0

From (i) and (ii) we get:

x² - 2A + G² = 0

Thus, x² - 2A + G² = 0 is the required quadratic equation.

Hence, x² - 2A + G² = 0 is the required quadratic equation.