Question

If AM and GM of the roots of a quadratic equation are 10 and 8 respectively then obtain the quadratic equation.

Answer 1

To find: The quadratic equation. 

Given: 

(i) AM of roots of quadratic equation is 10 

(ii) GM of roots of quadratic equation is 8

Formula used: (i) Arithmetic mean between a and b  = \(\frac{a + b}{2}\)

(ii) Geometric mean between a and b = \(\sqrt{ab}\)

Let the roots be p and q 

Arithmetic mean of roots p and q = \(\frac{p+q}{2} = 10\)

\(\frac{p+q}{2} = 10\)

⇒ p + q = 20 = sum of roots … (i) 

Geometric mean of roots p and q = \(\sqrt{Pq}\) = 8

⇒ pq = 64 = product of roots … (ii) 

Quadratic equation = x² – (sum of roots)x + (product of roots) 

From equation (i) and (ii) Quadratic equation 

= x² – (20)x + (64) 

= x² –20x + 64 

x² –20x + 64