The set of values of `p in R` for which the equation `x^2 +px+ 1/4 (p+2)= 0` will have real roots is

Options

A. `[2,+oo)`
B. `(-oo, 2]`
C. `(-oo,-1]`
D. `R-(-1,2)`

Answer

To find the set of values of \( p \in \mathbb{R} \) for which the equation

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

has real roots, we will use the condition that the discriminant of the quadratic equation must be greater than or equal to zero.

### Step 1: Identify coefficients
In the given quadratic equation, we can identify the coefficients:
- \( a = 1 \)
- \( b = p \)
- \( c = \frac{1}{4}(p + 2) \)

### Step 2: Write 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 \( c \):

\[ D = p^2 - 4 \cdot 1 \cdot \frac{1}{4}(p + 2) \]

### Step 3: Simplify the discriminant
Now, simplify the expression for the discriminant:

\[ D = p^2 - (p + 2) \]
\[ D = p^2 - p - 2 \]

### Step 4: Set the discriminant greater than or equal to zero
For the equation to have real roots, we need:

\[ p^2 - p - 2 \geq 0 \]

### Step 5: Factor the quadratic expression
Next, we factor the quadratic expression:

\[ p^2 - p - 2 = (p - 2)(p + 1) \]

### Step 6: Solve the inequality
We need to solve the inequality:

\[ (p - 2)(p + 1) \geq 0 \]

### Step 7: Find critical points
The critical points occur when each factor is zero:

1. \( p - 2 = 0 \) → \( p = 2 \)
2. \( p + 1 = 0 \) → \( p = -1 \)

### Step 8: Test intervals
We will test the sign of the expression \( (p - 2)(p + 1) \) in the intervals determined by the critical points \( -1 \) and \( 2 \):

1. **Interval \( (-\infty, -1) \)**: Choose \( p = -2 \)
\[ (-2 - 2)(-2 + 1) = (-4)(-1) = 4 \quad (\text{positive}) \]

2. **Interval \( (-1, 2) \)**: Choose \( p = 0 \)
\[ (0 - 2)(0 + 1) = (-2)(1) = -2 \quad (\text{negative}) \]

3. **Interval \( (2, \infty) \)**: Choose \( p = 3 \)
\[ (3 - 2)(3 + 1) = (1)(4) = 4 \quad (\text{positive}) \]

### Step 9: Write the solution set
From the test results, the expression \( (p - 2)(p + 1) \geq 0 \) is satisfied in the intervals:

- \( (-\infty, -1] \)
- \( [2, \infty) \)

Thus, the solution set for \( p \) is:

\[ p \in (-\infty, -1] \cup [2, \infty) \]

### Final Answer
The set of values of \( p \) for which the equation has real roots is:

\[ (-\infty, -1] \cup [2, \infty) \]