If quadratic equation `3x^(2)` - 4x + k = 0 has equal roots, then the value of k is............... .

Answer

To find the value of \( k \) for which the quadratic equation \( 3x^2 - 4x + k = 0 \) has equal roots, we need to use the concept of the discriminant. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

For the quadratic equation to have equal roots, the discriminant must be equal to zero:

\[ D = 0 \]

### Step-by-Step Solution:

1. **Identify coefficients**:
In the equation \( 3x^2 - 4x + k = 0 \), we identify:
- \( a = 3 \)
- \( b = -4 \)
- \( c = k \)

2. **Set up the discriminant**:
We set the discriminant equal to zero:
\[ D = b^2 - 4ac = 0 \]

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

4. **Calculate \( b^2 \)**:
Calculate \( (-4)^2 \):
\[ 16 - 12k = 0 \]

5. **Rearrange the equation**:
Rearranging gives:
\[ 16 = 12k \]

6. **Solve for \( k \)**:
Divide both sides by 12:
\[ k = \frac{16}{12} \]

7. **Simplify the fraction**:
Simplifying \( \frac{16}{12} \):
\[ k = \frac{4}{3} \]

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