Solve for k in 4x^2 + kx + 5 = 0 with Roots Having difference 1/4

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.

Explore how to solve the quadratic equation 4x^2 + kx + 5 = 0, where the roots differ by 1/4. Learn to apply Vieta's formulas for sum and product of roots to find the value of k. This problem is suited for high school students, particularly those in Grades 10-12.

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