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) \( x^2 - 10x + 16 = 0 \)
Let the roots be $\alpha$ and $\beta$.
Then: A.M. $= \frac{\alpha + \beta}{2} = 5 \implies \alpha + \beta = 10$. G.M. $= \sqrt{\alpha\beta} = 4 \implies \alpha\beta = 16$.
The quadratic equation is $x^2 - (\alpha + \beta)x + \alpha\beta = 0$, i.e., $x^2 - 10x + 16 = 0$.