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The question provides a quadratic inequality \( k s^2 - 4s > -\frac{1}{2} \), where \( k \) is a constant, and it specifies that the quadratic has repeat roots.

### Part (a)
We are asked to determine one possible value for \( k \), given that the quadratic possesses repeat roots. A quadratic has repeat roots when the discriminant \( \Delta \) of the quadratic is zero. 

The quadratic equation we are working with is:
\[
k s^2 - 4s + \frac{1}{2} = 0
\]
We rewrite it in the standard quadratic form \( A s^2 + B s + C = 0 \), where:
\[
A = k, \quad B = -4, \quad C = \frac{1}{2}
\]
The discriminant \( \Delta \) for a quadratic equation \( A s^2 + B s + C = 0 \) is given by:
\[
\Delta = B^2 - 4AC
\]
Substitute the values of \( A \), \( B \), and \( C \):
\[
\Delta = (-4)^2 - 4(k)(\frac{1}{2}) = 16 - 2k
\]
For repeat roots, we set the discriminant equal to zero:
\[
16 - 2k = 0
\]
Solving for \( k \):
\[
2k = 16 \quad \Rightarrow \quad k = 8
\]

So, one possible value for \( k \) is \( k = 8 \).

### Part (b)
We are now asked to solve the inequality \( k s^2 - 4s > -\frac{1}{2} \), using the value \( k = 8 \) from part (a).

The inequality becomes:
\[
8 s^2 - 4s > -\frac{1}{2}
\]
First, move all terms to one side:
\[
8 s^2 - 4s + \frac{1}{2} > 0
\]
Multiply through by 2 to eliminate the fraction:
\[
16 s^2 - 8s + 1 > 0
\]
This is now a quadratic inequality. The corresponding quadratic equation is:
\[
16 s^2 - 8s + 1 = 0
\]
Since this quadratic has repeat roots, the discriminant is zero. The root of the equation is:
\[
s = \frac{-(-8)}{2(16)} = \frac{8}{32} = \frac{1}{4}
\]

Thus, the quadratic has a double root at \( s = \frac{1}{4} \).

For the inequality \( 16 s^2 - 8s + 1 > 0 \), since the parabola opens upwards (because the coefficient of \( s^2 \) is positive), the quadratic will be positive for all \( s \) except at the double root \( s = \frac{1}{4} \), where it equals zero. Therefore, the solution to the inequality is:
\[
s \neq \frac{1}{4}
\]
In interval notation, this can be written as:
\[
s \in (-\infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty)
\]

Would you like a further explanation on any step? Here are five related questions to consider:
1. What happens to the solution if the quadratic had distinct roots instead of repeat roots?
2. How does the discriminant help us understand the nature of the roots of a quadratic?
3. How would the solution change if the sign in the inequality was \( \geq \) instead of \( > \)?
4. What would the graph of the quadratic equation \( 16s^2 - 8s + 1 = 0 \) look like?
5. Can you solve a similar inequality for different values of \( k \)?

**Tip:** Always check the discriminant to understand the nature of the roots when solving quadratic equations.

Given the quadratic inequality ks^2 - 4s > -1/2, where k is a constant and the quadratic possesses repeat roots. a) Determine one possible value for k. b) Using your answer from part a, solve the inequality.

Learn how to solve the quadratic inequality ks^2 - 4s > -1/2, where the quadratic has repeat roots. This problem involves using the discriminant to determine the value of k and solving the inequality step-by-step. Suitable for high school students studying algebra.

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