If p is chosen at random from the interval `[0, 6]` then the probability that the roots of the equation `x^2 +px+ 1/4 (p+2)= 0` will be real is

Options

A. `3/5`
B. `1/2`
C. `5/7`
D. `2/3`

Answer

To find the probability that the roots of the equation \( x^2 + px + \frac{1}{4}(p + 2) = 0 \) are real, we need to analyze the discriminant of the quadratic equation. The roots of the quadratic equation are real if the discriminant is non-negative.

### Step-by-step Solution:

1. **Identify the coefficients**:
The given quadratic equation can be rewritten as:
\[ x^2 + px + \frac{1}{4}(p + 2) = 0 \]
Here, \( A = 1 \), \( B = p \), and \( C = \frac{1}{4}(p + 2) \).

2. **Calculate the discriminant**:
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = p^2 - 4 \cdot 1 \cdot \frac{1}{4}(p + 2) \]
Simplifying this gives:
\[ D = p^2 - (p + 2) = p^2 - p - 2 \]

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

4. **Factor the quadratic expression**:
We can factor \( p^2 - p - 2 \):
\[ p^2 - p - 2 = (p - 2)(p + 1) \]
Thus, we need:
\[ (p - 2)(p + 1) \geq 0 \]

5. **Find the critical points**:
The critical points are \( p = 2 \) and \( p = -1 \). We will analyze the sign of the expression in the intervals defined by these points.

6. **Test intervals**:
- For \( p < -1 \): Choose \( p = -2 \):
\[ (-2 - 2)(-2 + 1) = (-4)(-1) = 4 \quad (\text{positive}) \]
- For \( -1 < p < 2 \): Choose \( p = 0 \):
\[ (0 - 2)(0 + 1) = (-2)(1) = -2 \quad (\text{negative}) \]
- For \( p > 2 \): Choose \( p = 3 \):
\[ (3 - 2)(3 + 1) = (1)(4) = 4 \quad (\text{positive}) \]

Thus, the solution to \( (p - 2)(p + 1) \geq 0 \) is:
\[ p \in (-\infty, -1] \cup [2, \infty) \]

7. **Consider the interval for \( p \)**:
Since \( p \) is chosen from the interval \( [0, 6] \), we only consider the interval \( [2, 6] \).

8. **Determine the sample space and the event**:
The sample space \( S \) is the interval \( [0, 6] \), which includes the values \( 0, 1, 2, 3, 4, 5, 6 \) (7 values).
The event \( E \) where the roots are real corresponds to the interval \( [2, 6] \), which includes the values \( 2, 3, 4, 5, 6 \) (5 values).

9. **Calculate the probability**:
The probability \( P \) that the roots are real is given by:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{7} \]

### Final Answer:
The probability that the roots of the equation will be real is \( \frac{5}{7} \).