If the equation `4x^2+x(p+1)+1=0` has exactly two equal roots, one of the value of p is

Options

A. 3
B. 0
C. -3
D. 5

Answer

To find the value of \( p \) for which the equation \( 4x^2 + x(p+1) + 1 = 0 \) has exactly two equal roots, we need to ensure that the discriminant of the quadratic equation is equal to zero.

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation**:
The given equation is \( 4x^2 + x(p+1) + 1 = 0 \).
Here, the coefficients are:
- \( a = 4 \)
- \( b = p + 1 \)
- \( c = 1 \)

2. **Write the formula for the discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For the roots to be equal, we need \( D = 0 \).

3. **Substitute the coefficients into the discriminant formula**:
Substitute \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (p + 1)^2 - 4 \cdot 4 \cdot 1 \]
Simplifying this gives:
\[ D = (p + 1)^2 - 16 \]

4. **Set the discriminant equal to zero**:
To find the value of \( p \) for which the roots are equal, set the discriminant to zero:
\[ (p + 1)^2 - 16 = 0 \]
Rearranging this gives:
\[ (p + 1)^2 = 16 \]

5. **Solve for \( p + 1 \)**:
Taking the square root of both sides, we have:
\[ p + 1 = 4 \quad \text{or} \quad p + 1 = -4 \]

6. **Find the values of \( p \)**:
From \( p + 1 = 4 \):
\[ p = 4 - 1 = 3 \]
From \( p + 1 = -4 \):
\[ p = -4 - 1 = -5 \]

7. **Conclusion**:
One of the values of \( p \) for which the equation has exactly two equal roots is \( p = 3 \) or \( p = -5 \).