The problem you have shared seems to be a quadratic equation where we need to find the value of $k$. Let's break it down:

The equation is in the form:

$x^2 + kx - \frac{5}{4} = 0$

It also states that one of the roots of this equation is $\frac{1}{2}$.

We can use the fact that if $r_1$ and $r_2$ are the roots of the quadratic equation $ax^2 + bx + c = 0$, then by Vieta's formulas:

1. The sum of the roots, $r_1 + r_2$, is given by $-\frac{b}{a}$.
2. The product of the roots, $r_1 \times r_2$, is given by $\frac{c}{a}$.

Here, we have:

• $a = 1$
• $b = k$
• $c = -\frac{5}{4}$
• One of the roots $r_1 = \frac{1}{2}$

From the product of the roots, we know that:

$r_1 \times r_2 = \frac{c}{a}$

So:

$\frac{1}{2} \times r_2 = -\frac{5}{4}$

Solving this for $r_2$:

$r_2 = -\frac{5}{4} \times \frac{2}{1} = -\frac{5}{2}$

Now, using the sum of the roots formula:

$r_1 + r_2 = -\frac{b}{a}$

Substituting the values:

$\frac{1}{2} + \left( -\frac{5}{2} \right) = -k$

Simplifying:

$\frac{1}{2} - \frac{5}{2} = -k$

$-\frac{4}{2} = -k$

$k = 2$

So, the value of $k$ is 2.

Would you like more details on any part of this explanation? Let me know if you have any further questions!

Here are some related questions:

1. How do we use the quadratic formula to find roots of a quadratic equation?
2. What are the applications of Vieta's formulas in solving quadratic equations?
3. Can we apply the same method to find the value of a different unknown in a quadratic equation?
4. How would the result change if the root were a different fraction?
5. What happens if the given root is negative?

Tip: Always check the conditions of the problem carefully, as small changes in the equation or roots can lead to very different solutions.