for what value of k, so that the equation 4x^2+kx+5=0 has roots whose difference is 1/4

To solve for the value of \( k \) such that the quadratic equation \( 4x^2 + kx + 5 = 0 \) has roots whose difference is \( \frac{1}{4} \), let's use the following properties of quadratic equations:

1. For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha \) and \( \beta \) is given by:
   \[
   \alpha + \beta = -\frac{b}{a}
   \]
   and the product of the roots is:
   \[
   \alpha \beta = \frac{c}{a}
   \]

2. Let the roots be \( \alpha \) and \( \beta \), and we are given that \( \alpha - \beta = \frac{1}{4} \).

### Step 1: Express the roots in terms of their sum and difference
We can express the roots \( \alpha \) and \( \beta \) as:
\[
\alpha = \frac{S + D}{2}, \quad \beta = \frac{S - D}{2}
\]
where \( S \) is the sum of the roots, and \( D \) is the difference between the roots. We are given \( D = \frac{1}{4} \).

### Step 2: Use the properties of the quadratic equation
For the equation \( 4x^2 + kx + 5 = 0 \), we have \( a = 4 \), \( b = k \), and \( c = 5 \). The sum and product of the roots are:
\[
\alpha + \beta = -\frac{k}{4} \quad \text{(sum of the roots)}
\]
\[
\alpha \beta = \frac{5}{4} \quad \text{(product of the roots)}
\]

### Step 3: Solve for \( S \) and \( k \)
From the first equation, we know that \( \alpha - \beta = \frac{1}{4} \). Therefore, the difference between the roots is given by:
\[
\alpha - \beta = \frac{1}{4}
\]

Now, using the identity \( (\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha \beta \), substitute the known values:
\[
\left(\frac{1}{4}\right)^2 = \left(-\frac{k}{4}\right)^2 - 4\left(\frac{5}{4}\right)
\]

Simplifying:
\[
\frac{1}{16} = \frac{k^2}{16} - 5
\]

Multiply through by 16 to eliminate the denominator:
\[
1 = k^2 - 80
\]

Solve for \( k^2 \):
\[
k^2 = 81
\]

Thus, \( k = \pm 9 \).

### Final Answer:
The value of \( k \) is \( \boxed{9} \) or \( \boxed{-9} \).

Would you like further details on any steps or have any other questions?

Here are some related questions you may find interesting:
1. What is the relationship between the discriminant and the difference of roots?
2. Can we find the roots explicitly for \( k = 9 \) and \( k = -9 \)?
3. What are other methods to solve quadratic equations with specific root properties?
4. How does changing the coefficient \( k \) affect the graph of the quadratic?
5. How can we derive the quadratic formula from first principles?

**Tip:** Whenever you are given specific properties of roots (like their sum, product, or difference), use those directly in the relationships derived from the quadratic equation.

For what value of k, so that the equation 4x^2 + kx + 5 = 0 has roots whose difference is 1/4?

Find the value of k in the quadratic equation 4x^2 + kx + 5 = 0, where the roots differ by 1/4. This problem involves key concepts from algebra and quadratic equations, using formulas for the sum and product of the roots, and is suitable for students in Grades 10-12.

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