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

Options

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

Answer

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

### Step 1: Substitute the zero into the polynomial
Since \( x - \frac{1}{2} \) is a zero, we can substitute \( x = \frac{1}{2} \) into the polynomial \( p(x) = 3x^2 + 2kx - 3 \) and set it equal to zero.

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

### Step 2: Calculate \( p\left(\frac{1}{2}\right) \)
Now, we will calculate \( p\left(\frac{1}{2}\right) \):

\[ p\left(\frac{1}{2}\right) = 3 \cdot \frac{1}{4} + 2k \cdot \frac{1}{2} - 3 \]

This simplifies to:

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

### Step 3: Set the equation to zero
Now, we can set the equation equal to zero:

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

### Step 4: Combine like terms
To combine the terms, we can convert \( -3 \) into a fraction with a denominator of 4:

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

Thus, the equation becomes:

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

This simplifies to:

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

### Step 5: Solve for \( k \)
Now, we can solve for \( k \):

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

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

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