The question shown in the image is:

Find the value of $p$, if $\frac{1}{2}$ is a root of the quadratic equation: $x^2 + px - \frac{5}{4} = 0$

Solution:

Since $\frac{1}{2}$ is a root of the quadratic equation, it must satisfy the equation. Therefore, substitute $x = \frac{1}{2}$ into the equation:

$\left(\frac{1}{2}\right)^2 + p\left(\frac{1}{2}\right) - \frac{5}{4} = 0$

1. Calculate $\left(\frac{1}{2}\right)^2$:

$\left(\frac{1}{2}\right)^2 = \frac{1}{4}$

2. Substitute into the equation:

$\frac{1}{4} + \frac{p}{2} - \frac{5}{4} = 0$

3. Combine like terms:

$\frac{1}{4} - \frac{5}{4} + \frac{p}{2} = 0$

$-\frac{4}{4} + \frac{p}{2} = 0$

$-1 + \frac{p}{2} = 0$

4. Solve for $p$:

$\frac{p}{2} = 1$

$p = 2$

Final Answer: $p = 2$

Would you like more details or have any further questions?

Here are 5 related questions to expand on this topic:

1. How do you find the other root of the quadratic equation after solving for $p$?
2. Can we derive the quadratic formula from this equation?
3. What are the possible values of $p$ if the equation has complex roots?
4. How do we determine whether the quadratic equation has real or imaginary roots?
5. How can you solve a quadratic equation using the factorization method?

Tip: When solving quadratic equations, always check whether the equation can be factored before using more complex methods like completing the square or the quadratic formula.