To complete the square for the quadratic function:

$y = x^2 - 6x + 1$

we'll rewrite it in the form $(x - h)^2 + k$, where $h$ and $k$ are constants.

Step-by-Step Solution:

1. Identify the coefficient of $x$: The coefficient of $x$ is $-6$.

2. Divide by 2 and square it: Take half of $-6$, which is $-3$, and then square it: $(-3)^2 = 9$

3. Add and subtract 9 inside the equation: To complete the square, add and subtract 9 within the equation: $y = x^2 - 6x + 9 - 9 + 1$

4. Rewrite as a perfect square: Now, the expression $x^2 - 6x + 9$ is a perfect square, so we can write it as: $y = (x - 3)^2 - 9 + 1$

5. Simplify the constants: Combine $-9$ and $+1$: $y = (x - 3)^2 - 8$

Final Answer:

The equation in completed square form is: $y = (x - 3)^2 - 8$

This shows that the vertex of the parabola is at $(3, -8)$.

Would you like more details or have any questions?

Here are some related questions to explore this topic further:

1. How do we find the vertex of a parabola directly from the equation?
2. What is the axis of symmetry for the parabola $y = x^2 - 6x + 1$?
3. How does completing the square help in solving quadratic equations?
4. What is the geometric interpretation of completing the square?
5. How does the form ((x - h)^2 + k\