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 \( x^2 - 3x + k = 0 \) such that it has equal roots, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -3 \) (coefficient of \( x \))
- \( c = k \) (constant term)

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

3. **Substitute the values into the discriminant formula**:
Substitute \( a \), \( b \), and \( c \) into the discriminant formula:
\[ (-3)^2 - 4 \cdot 1 \cdot k = 0 \]

4. **Simplify the equation**:
Calculate \( (-3)^2 \):
\[ 9 - 4k = 0 \]

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

### Final Answer:
The value of \( k \) is \( \frac{9}{4} \).

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