If the sum of the zeros of the polynomial `2x^(2)+3kx+3` is 6, then find the value of k.

Answer

To find the value of \( k \) given that the sum of the zeros of the polynomial \( 2x^2 + 3kx + 3 \) is 6, we can follow these steps:

### Step 1: Identify the coefficients of the polynomial
The polynomial is given as:
\[ 2x^2 + 3kx + 3 \]
Here, we can identify:
- \( a = 2 \)
- \( b = 3k \)
- \( c = 3 \)

### Step 2: Use the formula for the sum of the zeros
The sum of the zeros \( \alpha + \beta \) of a quadratic polynomial \( ax^2 + bx + c \) can be calculated using the formula:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values of \( a \) and \( b \):
\[ \alpha + \beta = -\frac{3k}{2} \]

### Step 3: Set the sum of the zeros equal to 6
According to the problem, the sum of the zeros is 6:
\[ -\frac{3k}{2} = 6 \]

### Step 4: Solve for \( k \)
To eliminate the fraction, multiply both sides of the equation by -2:
\[ 3k = -12 \]
Now, divide both sides by 3:
\[ k = -4 \]

### Final Answer
The value of \( k \) is:
\[ k = -4 \]

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