If an integer p is chosen at random in the interval `0le ple5,` then the probality that the roots of the equation `x^(2)+px+(p)/(4)+(1)/(2)=0` are real is -

Options

A. `1//5`
B. `2//5`
C. `3//5`
D. `4//5`

Answer

To solve the problem, we need to determine the probability that the roots of the quadratic equation

\[ x^2 + px + \left(\frac{p}{4} + \frac{1}{2}\right) = 0 \]

are real. For the roots to be real, the discriminant of the quadratic equation must be greater than or equal to zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The coefficients of the quadratic equation are:
- \( a = 1 \)
- \( b = p \)
- \( c = \frac{p}{4} + \frac{1}{2} \)

2. **Write the discriminant**:
The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = p^2 - 4 \left(1\right) \left(\frac{p}{4} + \frac{1}{2}\right) \]

3. **Simplify the discriminant**:
Expanding the expression:
\[ D = p^2 - 4\left(\frac{p}{4} + \frac{1}{2}\right) = p^2 - (4 \cdot \frac{p}{4}) - (4 \cdot \frac{1}{2}) = p^2 - p - 2 \]

4. **Set the discriminant greater than or equal to zero**:
For the roots to be real:
\[ p^2 - p - 2 \geq 0 \]

5. **Factor the quadratic**:
We can factor the quadratic:
\[ (p - 2)(p + 1) \geq 0 \]

6. **Determine the intervals**:
The critical points are \( p = 2 \) and \( p = -1 \). We analyze the sign of the expression in the intervals:
- \( (-\infty, -1) \)
- \( (-1, 2) \)
- \( (2, \infty) \)

Testing values from each interval:
- For \( p < -1 \), say \( p = -2 \): \( (-2 - 2)(-2 + 1) = (-4)(-1) > 0 \)
- For \( -1 < p < 2 \), say \( p = 0 \): \( (0 - 2)(0 + 1) = (-2)(1) < 0 \)
- For \( p > 2 \), say \( p = 3 \): \( (3 - 2)(3 + 1) = (1)(4) > 0 \)

Thus, the solution to the inequality is:
\[ p \in (-\infty, -1] \cup [2, \infty) \]

7. **Consider the range of p**:
Since \( p \) is chosen from the interval \( [0, 5] \), we only consider:
\[ p \in [2, 5] \]

8. **Count the valid integer values**:
The integers in the range \( [2, 5] \) are \( 2, 3, 4, 5 \). This gives us 4 valid integers.

9. **Total possible integers in the range [0, 5]**:
The integers in the interval \( [0, 5] \) are \( 0, 1, 2, 3, 4, 5 \), which gives us 6 total integers.

10. **Calculate the probability**:
The probability \( P \) that the roots are real is:
\[ P = \frac{\text{Number of valid integers}}{\text{Total integers}} = \frac{4}{6} = \frac{2}{3} \]

### Final Answer:
The probability that the roots of the equation are real is \( \frac{2}{3} \).