If `x=-(1)/(2)` is a solution of the quadratic equation `3x^(2)+2kx-3=0`, find the velue of k.

Answer

To find the value of \( k \) in the quadratic equation \( 3x^2 + 2kx - 3 = 0 \) given that \( x = -\frac{1}{2} \) is a solution, we will substitute \( x \) into the equation and solve for \( k \).

### Step-by-Step Solution:

1. **Substitute \( x = -\frac{1}{2} \) into the equation:**
\[ 3\left(-\frac{1}{2}\right)^2 + 2k\left(-\frac{1}{2}\right) - 3 = 0 \]

2. **Calculate \( \left(-\frac{1}{2}\right)^2 \):**
\[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]
Therefore, substituting this into the equation gives:
\[ 3 \cdot \frac{1}{4} + 2k\left(-\frac{1}{2}\right) - 3 = 0 \]

3. **Multiply \( 3 \cdot \frac{1}{4} \):**
\[ \frac{3}{4} + 2k\left(-\frac{1}{2}\right) - 3 = 0 \]
This simplifies to:
\[ \frac{3}{4} - k - 3 = 0 \]

4. **Combine the constant terms:**
\[ \frac{3}{4} - 3 = \frac{3}{4} - \frac{12}{4} = -\frac{9}{4} \]
Thus, the equation now is:
\[ -k - \frac{9}{4} = 0 \]

5. **Solve for \( k \):**
\[ -k = \frac{9}{4} \]
Therefore, multiplying both sides by -1 gives:
\[ k = -\frac{9}{4} \]

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

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