For what value of k the quadratic equation `x ^(2) - kx +4` have real and equal roots ?

Answer

To find the value of \( k \) for which the quadratic equation \( x^2 - kx + 4 \) has real and equal roots, we need to use the condition for real and equal roots of a quadratic equation. This condition is that the discriminant must be equal to zero.

### Step-by-Step Solution:

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

2. **Write the discriminant**:
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]

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

4. **Set the discriminant equal to zero**:
For the roots to be real and equal, we set the discriminant \( D \) to zero:
\[ k^2 - 16 = 0 \]

5. **Solve for \( k \)**:
Rearranging the equation gives:
\[ k^2 = 16 \]
Taking the square root of both sides, we find:
\[ k = \pm 4 \]

### Final Answer:
The values of \( k \) for which the quadratic equation \( x^2 - kx + 4 \) has real and equal roots are \( k = 4 \) and \( k = -4 \).

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