To convert the quadratic equation \( 2x^2 + 4x + 6 \) into vertex form, we use the process called **completing the square**. Here's the step-by-step process:

### Given equation:
\[
y = 2x^2 + 4x + 6
\]

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

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

### Step 2: Complete the square
Now, focus on the expression inside the parentheses: \(x^2 + 2x\). To complete the square, take half of the coefficient of \(x\) (which is 2), square it, and add it inside the parentheses. Half of 2 is 1, and \(1^2 = 1\).

So, we add and subtract 1 inside the parentheses to maintain equality:

\[
y = 2(x^2 + 2x + 1 - 1) + 6
\]
\[
y = 2((x + 1)^2 - 1) + 6
\]

### Step 3: Simplify
Now distribute the 2 and simplify the expression:

\[
y = 2(x + 1)^2 - 2 + 6
\]
\[
y = 2(x + 1)^2 + 4
\]

### Final result (vertex form):
The vertex form of the equation is:

\[
y = 2(x + 1)^2 + 4
\]

In this form, the vertex is at \( (-1, 4) \).

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Would you like more details or have any questions?

Here are five related questions for further practice:
1. Convert \( 3x^2 + 6x + 5 \) to vertex form.
2. Find the vertex of the equation \( y = -x^2 + 4x - 7 \).
3. What is the axis of symmetry for \( y = 2(x + 1)^2 + 4 \)?
4. Convert \( -2x^2 + 8x - 3 \) to vertex form.
5. How does completing the square help in solving quadratic equations?

**Tip**: The vertex form is useful because it immediately reveals the vertex of the parabola, making it easier to analyze its graph.

Learn how to convert the quadratic equation 2x^2 + 4x + 6 into vertex form using the completing the square method. This solution includes a step-by-step explanation, revealing key concepts in algebra and quadratic equations, ideal for students in Grades 8-10.

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