If the roots of polynomial `x^2-3x+k=0` has equal roots, find the value of k:

Options

A. `3/2`
B. `9/4`
C. `1/4`
D. `1/6`

Answer

To find the value of \( k \) for which the polynomial \( x^2 - 3x + k = 0 \) has equal roots, we can use the concept of the discriminant from the quadratic formula.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given polynomial is \( x^2 - 3x + k = 0 \). Here, the coefficients are:
- \( a = 1 \)
- \( b = -3 \)
- \( c = k \)

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

3. **Substitute the coefficients into the discriminant formula**:
\[ D = (-3)^2 - 4 \cdot 1 \cdot k \]
Simplifying this gives:
\[ D = 9 - 4k \]

4. **Set the discriminant equal to zero**:
Since we want the roots to be equal, we set the discriminant \( D \) to zero:
\[ 9 - 4k = 0 \]

5. **Solve for \( k \)**:
Rearranging the equation gives:
\[ 4k = 9 \]
Dividing both sides by 4:
\[ k = \frac{9}{4} \]

### Final Answer:
The value of \( k \) for which the polynomial \( x^2 - 3x + k = 0 \) has equal roots is \( k = \frac{9}{4} \).