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!

Learn how to rewrite the quadratic equation x^2 + 6x + 2 in vertex form by completing the square. This tutorial covers algebraic concepts, such as vertex form and completing the square, with clear steps and examples. Ideal for students in Grades 8-10.

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