If the quadratic equation `4 x^ 2 - (p-2)x+ 1 = 0` has equal roots, then the value of 'p' are

Options

A. `2 or 6 `
B. `2 or - 6`
C. `-2 or - 6`
D. `-2 or 6`

Answer

To find the value of \( p \) for which the quadratic equation \( 4x^2 - (p-2)x + 1 = 0 \) has equal roots, we need to use the condition for equal roots in a quadratic equation. The condition is that the discriminant must be zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, \( a = 4 \), \( b = -(p-2) \), and \( c = 1 \).

2. **Write the discriminant condition**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
For the roots to be equal, we set the discriminant to zero:
\[ D = 0 \]

3. **Substitute the coefficients into the discriminant**:
Substitute \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (-(p-2))^2 - 4 \cdot 4 \cdot 1 = 0 \]
Simplifying this gives:
\[ (p-2)^2 - 16 = 0 \]

4. **Solve the equation**:
Rearranging the equation, we have:
\[ (p-2)^2 = 16 \]
Taking the square root of both sides, we get:
\[ p - 2 = 4 \quad \text{or} \quad p - 2 = -4 \]

5. **Find the values of \( p \)**:
Solving these two equations:
- From \( p - 2 = 4 \):
\[ p = 4 + 2 = 6 \]
- From \( p - 2 = -4 \):
\[ p = -4 + 2 = -2 \]

6. **Conclusion**:
The values of \( p \) for which the quadratic equation has equal roots are:
\[ p = 6 \quad \text{and} \quad p = -2 \]

### Final Answer:
The values of \( p \) are \( 6 \) and \( -2 \).