To rewrite the quadratic equation $x^2 + 6x + 2$ in vertex form, we use the method of completing the square.

The vertex form of a quadratic equation is:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola.

Let's go through the steps:

1. Start with the original equation:

$y = x^2 + 6x + 2$

2. Isolate the quadratic and linear terms:

$y = (x^2 + 6x) + 2$

3. Complete the square:

To complete the square, take the coefficient of $x$ (which is 6), divide it by 2, and square it:

$\left(\frac{6}{2}\right)^2 = 9$

Now, add and subtract 9 inside the parentheses:

$y = (x^2 + 6x + 9 - 9) + 2$

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

4. Simplify:

The expression inside the parentheses is now a perfect square:

$y = (x + 3)^2 - 7$

Final answer:

The vertex form of the quadratic equation $x^2 + 6x + 2$ is:

$y = (x + 3)^2 - 7$

This shows that the vertex is at $(-3, -7)$.

Would you like a deeper explanation of any part of this process?

Here are 5 related questions to expand on this topic:

1. How do you find the vertex from a quadratic equation in standard form?
2. What does completing the square mean geometrically?
3. How can the discriminant help in understanding the roots of a quadratic equation?
4. How do you convert a quadratic equation to factored form?
5. What is the significance of the axis of symmetry in a parabola?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value!