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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.

Find the value of p, if 1/2 is a root of the quadratic equation x^2 + px - 5/4 = 0.

Learn how to solve for the value of p in the quadratic equation x^2 + px - 5/4 = 0, where 1/2 is a known root. This step-by-step solution covers key algebraic concepts and is suitable for students in Grades 10-12.

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