If AM and GM of the roots of a quadratic equation are 10 and 8 respect

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

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

Answer»

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