Question

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Answer 1

Let the root of the quadratic equation be a and b. 

According to the given condition,

⇒ AM = \(\frac{a+ b}{2} = A\)

⇒ a + b = 2A …..(1) 

⇒ GM = √ab = G 

= ab = G²…(2) 

The quadratic equation is given by, 

x²– x (Sum of roots) + (Product of roots) = 0 

x² – x (2A) + (G²) = 0 

x² – 2Ax + G² = 0 [Using (1) and (2)] 

Thus, the required quadratic equation is x² – 2Ax + G² = 0.