https://file.aiquickdraw.com/mathbot/file/1728596687867vmozhzo3.jpg

To rewrite the quadratic equation \( y = 2x^2 - 2x - 12 \) in the form \( a(x + h)^2 + k \), we need to complete the square. Here’s the step-by-step process:

### Step 1: Factor out the coefficient of \( x^2 \) from the first two terms
Factor out the 2 from the terms involving \( x \):

\[
y = 2(x^2 - x) - 12
\]

### Step 2: Complete the square inside the parentheses
To complete the square, take half of the coefficient of \( x \), which is \( -1 \), and square it:

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

Now, add and subtract \( \frac{1}{4} \) inside the parentheses:

\[
y = 2\left( x^2 - x + \frac{1}{4} - \frac{1}{4} \right) - 12
\]

### Step 3: Rewrite the perfect square trinomial
The expression \( x^2 - x + \frac{1}{4} \) is a perfect square trinomial, so we can rewrite it as:

\[
y = 2\left( \left( x - \frac{1}{2} \right)^2 - \frac{1}{4} \right) - 12
\]

### Step 4: Simplify the equation
Distribute the 2:

\[
y = 2\left( x - \frac{1}{2} \right)^2 - 2\left( \frac{1}{4} \right) - 12
\]

\[
y = 2\left( x - \frac{1}{2} \right)^2 - \frac{1}{2} - 12
\]

Finally, combine constants:

\[
y = 2\left( x - \frac{1}{2} \right)^2 - 12.5
\]

### Final Answer
The quadratic equation in the form \( a(x + h)^2 + k \) is:

\[
y = 2\left( x - \frac{1}{2} \right)^2 - 12.5
\]

---

Would you like more details on any part of the solution or have any further questions?

Here are 5 related questions to explore:

1. How do you find the vertex of a quadratic equation from the completed square form?
2. What is the significance of completing the square in solving quadratic equations?
3. How can we graph a quadratic function given its vertex form?
4. Can you explain how to solve quadratic equations using the quadratic formula?
5. How does the value of \( a \) in the equation \( y = a(x + h)^2 + k \) affect the shape of the parabola?

**Tip:** Completing the square is also a helpful technique in integrating certain functions in calculus!

Given the quadratic equation y = 2x^2 - 2x - 12, write the equation in the form a(x + h)^2 + k.

Learn how to rewrite the quadratic equation y = 2x^2 - 2x - 12 into vertex form using the completing the square method. This step-by-step solution explains key algebraic concepts and techniques. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation