Given the quadratic equation $4x^2 - 9x - 16 = 0$, where $\alpha$ and $\beta$ are its roots, we need to determine which of the following statements are correct: I. $\alpha + \beta = \frac{9}{4}$ II. $\alpha \beta = -4$ III. $\alpha + \beta = \frac{4}{9}$

1. Problem Description

Given the quadratic equation $4x^2 - 9x - 16 = 0$, where $\alpha$ and $\beta$ are its roots, we need to determine which of the following statements are correct:

I. $\alpha + \beta = \frac{9}{4}$

II. $\alpha \beta = -4$

III. $\alpha + \beta = \frac{4}{9}$

2. Solution Steps

For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is given by:

$\alpha + \beta = -\frac{b}{a}$

and the product of the roots is given by:

$\alpha \beta = \frac{c}{a}$

In our case, $a = 4$, $b = -9$, and $c = -16$.

Using the formula for the sum of the roots, we have:

$\alpha + \beta = -\frac{-9}{4} = \frac{9}{4}$

So, statement I is correct.

Using the formula for the product of the roots, we have:

$\alpha \beta = \frac{-16}{4} = -4$

So, statement II is correct.

Statement III states $\alpha + \beta = \frac{4}{9}$, which is incorrect because we already found that $\alpha + \beta = \frac{9}{4}$.

Therefore, statements I and II are correct.

3. Final Answer

C. I and II only