The function g is defined by `g(x)=2x^2-dx-6`, where d is a constant. If one of the zeros of g is 6, what is the value of the other zero of g?

Options

A. 2
B. `1/2`
C. `-1/2`
D. -2

Answer

To solve the problem, we need to find the other zero of the function \( g(x) = 2x^2 - dx - 6 \) given that one of the zeros is \( x = 6 \).

### Step-by-Step Solution:

1. **Substitute the known zero into the function**: Since \( g(6) = 0 \), we can substitute \( x = 6 \) into the function to find \( d \).
\[ g(6) = 2(6^2) - d(6) - 6 = 0 \]
This simplifies to:
\[ 2(36) - 6d - 6 = 0 \]
\[ 72 - 6d - 6 = 0 \]
\[ 66 - 6d = 0 \]

2. **Solve for \( d \)**: Rearranging the equation gives:
\[ 6d = 66 \]
Dividing both sides by 6:
\[ d = 11 \]

3. **Substitute \( d \) back into the function**: Now that we have \( d \), we can rewrite the function:
\[ g(x) = 2x^2 - 11x - 6 \]

4. **Use the quadratic formula to find the zeros**: The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -11 \), and \( c = -6 \).

5. **Calculate the discriminant**:
\[ b^2 - 4ac = (-11)^2 - 4(2)(-6) = 121 + 48 = 169 \]

6. **Substitute values into the quadratic formula**:
\[ x = \frac{-(-11) \pm \sqrt{169}}{2(2)} = \frac{11 \pm 13}{4} \]

7. **Calculate the two possible values for \( x \)**:
- First zero:
\[ x_1 = \frac{11 + 13}{4} = \frac{24}{4} = 6 \]
- Second zero:
\[ x_2 = \frac{11 - 13}{4} = \frac{-2}{4} = -\frac{1}{2} \]

8. **Conclusion**: The other zero of the function \( g(x) \) is:
\[ -\frac{1}{2} \]

### Final Answer:
The value of the other zero of \( g \) is \( -\frac{1}{2} \).