Question

If AM and GM of roots of a quadratic equation are 8 and 5 respectively. Find quadratic equation?
1. x² - 16x - 25 = 0
2. x² - 16x + 25 = 0
3. x² + 25x - 16 = 0
4. x² - 25x - 16 = 0

Answer 1

Correct Answer - Option 2 : x² - 16x + 25 = 0

Concept:

If A is the arithmetic mean of numbers a and b and is given by: \({\rm{A}} = {\rm{\;}}\frac{{{\rm{a\;}} + {\rm{\;b}}}}{2}\)

If G is the geometric mean of the numbers a and b and is given by: \({\rm{G}} = {\rm{\;}}\sqrt {{\rm{ab}}} \)

Calculation:

Let α and β be the two roots of the quadratic equation

Given:

The arithmetic mean of the roots of a quadratic equation is 8

Therefore, \(\frac{{{\rm{\alpha }} + {\rm{\beta }}}}{2} = 8\)

⇒ α + β = 16

Now, geometric mean of the roots of a quadratic equation is 5.

Therefore, \(\sqrt {{\rm{\alpha \beta }}} = 5\)

Squaring both sides, we get

⇒ αβ = 25     

Thus, the required equation is:

⇒ x2 - (Sum of the roots) x + product of the roots = 0

⇒ Thus, the required equation is:

⇒ x2 - (α + β) x + αβ = 0

⇒ x² -16x + 25 = 0