Question

If A.M. and G.M. of roots of a quadratic equation are $5$ and $4$ respectively, then the quadratic equation is:

• A. $x^2 - 10x - 16 = 0$
• B. $x^2 + 10x + 16 = 0$
• C. $x^2 + 10x - 16 = 0$
• D. $x^2 - 10x + 16 = 0$

Answer 1

The Correct Option is D

1. Understand the problem:

We are given the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of the roots of a quadratic equation as 5 and 4, respectively. We need to find the quadratic equation from the given options.

2. Let the roots be \( \alpha \) and \( \beta \):

The A.M. of the roots is given by:

\[ \frac{\alpha + \beta}{2} = 5 \implies \alpha + \beta = 10 \]

The G.M. of the roots is given by:

\[ \sqrt{\alpha \beta} = 4 \implies \alpha \beta = 16 \]

3. Form the quadratic equation:

A quadratic equation with roots \( \alpha \) and \( \beta \) is:

\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]

Substituting the values:

\[ x^2 - 10x + 16 = 0 \]

Correct Answer: (D) \( x^2 - 10x + 16 = 0 \)

Answer 2

Let the roots of the quadratic equation be \( \alpha \) and \( \beta \).

The arithmetic mean (AM) of the roots is given by:

\[ \text{AM} = \frac{\alpha + \beta}{2} = 5 \] \[ \alpha + \beta = 10 \]

The geometric mean (GM) of the roots is given by:

\[ \text{GM} = \sqrt{\alpha \beta} = 4 \] \[ \alpha \beta = 16 \]

A quadratic equation with roots \( \alpha \) and \( \beta \) can be written as:

\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]

Substituting the values of \( \alpha + \beta \) and \( \alpha \beta \), we get:

\[ x^2 - 10x + 16 = 0 \]

Therefore, the quadratic equation is \( x^2 - 10x + 16 = 0 \).