To convert the quadratic equation $y = x^2 + 12x$ into vertex form, we'll complete 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.

Steps:

1. Start with the given equation: $y = x^2 + 12x$

2. Complete the square by taking half of the coefficient of $x$ (which is 12), squaring it, and adding and subtracting that value inside the equation: $\text{Half of 12} = 6, \quad 6^2 = 36$ Add and subtract 36 inside the equation: $y = (x^2 + 12x + 36) - 36$

3. Rewrite the trinomial as a perfect square: $y = (x + 6)^2 - 36$

So, the vertex form of the equation is:

$y = (x + 6)^2 - 36$

Vertex:

The vertex is $(-6, -36)$.

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How can you determine the vertex from the standard form of a quadratic equation?
2. What does completing the square mean geometrically?
3. How do you convert a quadratic equation into factored form?
4. How do the values of $h$ and $k$ affect the graph of the parabola?
5. What is the axis of symmetry of the parabola, and how do you find it?

Tip: The vertex of a parabola in standard form $ax^2 + bx + c$ can also be found using the formula $x = -\frac{b}{2a}$.