Transform to its vertex form y=2x²+12x+15

To transform the quadratic equation \( y = 2x^2 + 12x + 15 \) into its vertex form, we will complete the square. Here's the step-by-step process:

### Step 1: Factor out the coefficient of \( x^2 \)
First, we factor out the 2 from the terms involving \( x \):

\[
y = 2(x^2 + 6x) + 15
\]

### Step 2: Complete the square
Now, we complete the square inside the parentheses. To complete the square, take half of the coefficient of \( x \) (which is 6), square it, and then add and subtract this value inside the parentheses. Half of 6 is 3, and squaring 3 gives 9.

\[
y = 2(x^2 + 6x + 9 - 9) + 15
\]

### Step 3: Simplify the expression
The expression inside the parentheses becomes a perfect square trinomial:

\[
y = 2((x + 3)^2 - 9) + 15
\]

Now, distribute the 2 across the terms inside the parentheses:

\[
y = 2(x + 3)^2 - 18 + 15
\]

### Step 4: Combine constants
Finally, combine the constants:

\[
y = 2(x + 3)^2 - 3
\]

### Vertex Form:
The vertex form of the quadratic equation is:

\[
y = 2(x + 3)^2 - 3
\]

The vertex of the parabola is at \( (-3, -3) \).

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Would you like more details or have any questions?

### Related Questions:
1. How do you find the vertex of a parabola from its standard form?
2. What is the significance of completing the square?
3. How do you graph a quadratic equation in vertex form?
4. Can the vertex form help in solving quadratic inequalities?
5. How do transformations affect the graph of a quadratic equation?

### Tip:
In vertex form \( y = a(x-h)^2 + k \), the vertex is always at \( (h, k) \).

Learn how to transform the quadratic equation y = 2x² + 12x + 15 into its vertex form using the method of completing the square. This step-by-step guide covers key algebraic concepts and is suitable for students in Grades 9-10.

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