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To find the zeros of the function \( f(x) = 2x^2 - 2x - 12 \), we need to solve the equation \( f(x) = 0 \). This leads to the quadratic equation:

\[
2x^2 - 2x - 12 = 0
\]

### Step 1: Simplify the equation
We can divide the entire equation by 2 to make it simpler:

\[
x^2 - x - 6 = 0
\]

### Step 2: Factor the quadratic equation
We now need to factor \( x^2 - x - 6 \). We are looking for two numbers that multiply to -6 and add up to -1. The numbers -3 and 2 satisfy this condition. So, we can factor the quadratic as:

\[
(x - 3)(x + 2) = 0
\]

### Step 3: Solve for the zeros
Now, we solve for \( x \) by setting each factor equal to zero:

1. \( x - 3 = 0 \)  ⟹  \( x = 3 \)
2. \( x + 2 = 0 \)  ⟹  \( x = -2 \)

### Conclusion:
The zeros of the function are \( x = 3 \) and \( x = -2 \).

Would you like further details or have any questions?

Here are 5 related questions to extend this topic:
1. How do you determine the factors of a quadratic equation if it cannot be factored easily?
2. What is the significance of the zeros of a function in graphing?
3. Can you find the vertex of the quadratic function \( f(x) = 2x^2 - 2x - 12 \)?
4. How would you solve the quadratic equation if factoring wasn't possible?
5. What is the relationship between the discriminant and the number of real zeros of a quadratic equation?

**Tip:** When solving quadratic equations, always check if the quadratic can be factored first, as this method is often quicker than using the quadratic formula.

Find the zeros of the function f(x) = 2x^2 - 2x - 12.

Learn how to solve the quadratic equation f(x) = 2x^2 - 2x - 12 to find its zeros by factoring. This step-by-step guide simplifies the process and highlights key algebraic concepts. Suitable for students in Grades 9-11.

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