What is the value of k in the quadratic is one of polynomial `3x^(2) + 2kx -3`, if `x = (1)/(2)` the zeroes of it?

Options

A. `(1)/(5)`
B. `(3)/(2)`
C. `-(1)/(4)`
D. `-(9)/(4)`

Answer

To find the value of \( k \) in the polynomial \( 3x^2 + 2kx - 3 \) given that \( x = \frac{1}{2} \) is one of its zeroes, we can follow these steps:

### Step 1: Substitute \( x = \frac{1}{2} \) into the polynomial
Since \( x = \frac{1}{2} \) is a zero of the polynomial, we can substitute this value into the polynomial and set it equal to zero:

\[ 3\left(\frac{1}{2}\right)^2 + 2k\left(\frac{1}{2}\right) - 3 = 0 \]

### Step 2: Simplify the equation
Now, calculate \( 3\left(\frac{1}{2}\right)^2 \):

\[ 3\left(\frac{1}{2}\right)^2 = 3 \cdot \frac{1}{4} = \frac{3}{4} \]

Next, calculate \( 2k\left(\frac{1}{2}\right) \):

\[ 2k\left(\frac{1}{2}\right) = k \]

Now substitute these values back into the equation:

\[ \frac{3}{4} + k - 3 = 0 \]

### Step 3: Combine like terms
Now, we need to combine the constants:

\[ k + \frac{3}{4} - 3 = 0 \]

To combine \( -3 \) with \( \frac{3}{4} \), convert \( -3 \) into a fraction with a denominator of 4:

\[ -3 = -\frac{12}{4} \]

So the equation becomes:

\[ k + \frac{3}{4} - \frac{12}{4} = 0 \]

Combine the fractions:

\[ k - \frac{9}{4} = 0 \]

### Step 4: Solve for \( k \)
Now, isolate \( k \):

\[ k = \frac{9}{4} \]

Thus, the value of \( k \) is \( \frac{9}{4} \).

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

---