Question

If A.M. and G.M. of the roots of a quadratic equation are 8 and 5 respectively, then the quadratic equation is

• A. $x^2 + 8x + 5 = 0$
• B. $x^2 - 16x + 10 = 0$
• C. $x^2 - 16x + 25 = 0$
• D. $x^2 + 8x + 25 = 0$
• E. $x^2 + 10x + 15 = 0$

Hint:

Always remember: A.M. gives sum of roots, and G.M. gives product of roots — perfect for forming quadratic equations quickly.

Answer 1

The Correct Option is C

Concept:
If $\alpha$ and $\beta$ are the roots of a quadratic equation: \[ x^2 - (\alpha + \beta)x + \alpha\beta = 0 \] Also,
• Arithmetic Mean (A.M.): $\frac{\alpha + \beta}{2}$
• Geometric Mean (G.M.): $\sqrt{\alpha\beta}$

Step 1: Use given A.M. to find sum of roots.
\[ \frac{\alpha + \beta}{2} = 8 \] \[ \alpha + \beta = 16 \]

Step 2: Use given G.M. to find product of roots.
\[ \sqrt{\alpha\beta} = 5 \] \[ \alpha\beta = 25 \]

Step 3: Form the quadratic equation.
\[ x^2 - (\alpha + \beta)x + \alpha\beta = 0 \] Substitute values: \[ x^2 - 16x + 25 = 0 \]
Final Answer: \[ \boxed{x^2 - 16x + 25 = 0} \]