Question

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

Answer 1

Let two roots be a and b

∴ The arithmetic mean is given by \(\frac{a+b}{2}\)

⇒ A = \(\frac{a+b}{2}\)

⇒ Geometric mean is given by √a.b

⇒ G = √ab

Quadratic equation can be written as

⇒ x² - (a+b)x+ab = 0

Where a and b are roots of given equation

Substituting AM and GM 

⇒ x² - 2Ax+G² = 0.