We are given the quadratic equation:

$4x^2 + kx + 5 = 0$

We need to find the value of $k$ such that the roots of this equation have a difference of $\frac{1}{4}$.

Let the roots of the quadratic equation be $\alpha$ and $\beta$. Using Vieta's formulas, we know that:

1. The sum of the roots: $\alpha + \beta = -\frac{k}{4}$
2. The product of the roots: $\alpha \beta = \frac{5}{4}$

We are also told that the difference between the roots is $\alpha - \beta = \frac{1}{4}$.

Step 1: Use sum and difference of roots

From the equations for sum and difference, we can express the roots as:

$\alpha = \frac{-\frac{k}{4} + \frac{1}{4}}{2}, \quad \beta = \frac{-\frac{k}{4} - \frac{1}{4}}{2}$

Step 2: Use the product of the roots

The product of the roots is given as:

$\alpha \beta = \left( \frac{-\frac{k}{4} + \frac{1}{4}}{2} \right) \times \left( \frac{-\frac{k}{4} - \frac{1}{4}}{2} \right)$

This equals $\frac{5}{4}$.

We can now solve for $k$.The value of $k$ that satisfies the condition of the roots having a difference of $\frac{1}{4}$ can be either:

$k = -9 \quad \text{or} \quad k = 9$

Would you like to see more details on how these values were derived, or any further clarifications?

Here are some related questions:

1. What is the significance of Vieta's formulas in solving quadratic equations?
2. How does the discriminant relate to the nature of the roots of a quadratic equation?
3. Can there be complex roots if $k = 9$ or $k = -9$?
4. How would you graphically interpret the solution with these values of $k$?
5. What happens to the quadratic equation if the difference between roots is changed?

Tip: When solving quadratic equations, using sum and product of roots formulas can simplify the process if specific root conditions are provided.