If `1/2` is a root of x ^(2) + px - 5/4 =0` then value of p is

Options

A. 2
B. `-2`
C. `1/4`
D. `1/2`

Answer

To find the value of \( p \) in the quadratic equation \( x^2 + px - \frac{5}{4} = 0 \) given that \( \frac{1}{2} \) is a root, we can follow these steps:

### Step 1: Substitute the root into the equation
Since \( \frac{1}{2} \) is a root of the equation, we can substitute \( x = \frac{1}{2} \) into the equation:

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

### Step 2: Calculate \( \left(\frac{1}{2}\right)^2 \)
Calculating \( \left(\frac{1}{2}\right)^2 \):

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

### Step 3: Substitute back into the equation
Now substituting this value back into the equation gives us:

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

### Step 4: Simplify the equation
Next, we simplify the equation:

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

This simplifies to:

\[ -\frac{4}{4} + \frac{p}{2} = 0 \]

Which further simplifies to:

\[ -1 + \frac{p}{2} = 0 \]

### Step 5: Solve for \( p \)
Now, we can isolate \( p \):

\[ \frac{p}{2} = 1 \]

Multiplying both sides by 2 gives:

\[ p = 2 \]

### Final Answer
Thus, the value of \( p \) is:

\[ \boxed{2} \]