If the quadratic equation `x^2-3x+k=0` has equal roots, then what is the value of k?

Options

A. `(-4)/9`
B. `(-9)/4`
C. `9/4`
D. `(-4)/9`

Answer

To find the value of \( k \) for the quadratic equation \( x^2 - 3x + k = 0 \) to have equal roots, we need to use the condition for equal roots in a quadratic equation.

### Step-by-step Solution:

1. **Identify the standard form of a quadratic equation**:
The standard form of a quadratic equation is given by \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -3 \), and \( c = k \).

2. **Condition for equal roots**:
A quadratic equation has equal roots if the discriminant \( D \) is equal to zero. The discriminant is given by the formula:
\[ D = b^2 - 4ac \]
For our equation, substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-3)^2 - 4 \cdot 1 \cdot k \]

3. **Calculate the discriminant**:
\[ D = 9 - 4k \]

4. **Set the discriminant to zero**:
For 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 \( k \) is \( \frac{9}{4} \).