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.