To rewrite the quadratic $y = x^2 + 10x + 28$ in vertex form and determine the vertex coordinates, we complete the square.

Steps:

1. Group the $x^2$ and $x$ terms: $y = (x^2 + 10x) + 28$

2. Complete the square:

   • Take half of the coefficient of $x$ (which is 10), square it, and add it inside the parentheses: $\text{Half of } 10 = 5, \quad 5^2 = 25$
   • Add and subtract 25 inside the parentheses to keep the equation balanced: $y = (x^2 + 10x + 25 - 25) + 28$

3. Rewrite as a perfect square trinomial:

   • The terms $x^2 + 10x + 25$ factor as $(x + 5)^2$: $y = (x + 5)^2 - 25 + 28$

4. Simplify: $y = (x + 5)^2 + 3$

Vertex Form:

$y = (x + 5)^2 + 3$

Vertex:

The vertex is $(-5, 3)$.

Would you like a detailed explanation of any specific step? Here are some follow-up questions you might find helpful:

1. How does completing the square work in general?
2. Why do we subtract and add the same value when completing the square?
3. Can vertex form help identify the direction of the parabola?
4. How does the vertex form relate to the graph of the function?
5. Can we confirm the vertex coordinates using the derivative?

Tip: Completing the square is a useful technique not just for rewriting quadratics but also for solving quadratic equations!