https://file.aiquickdraw.com/mathbot/file/17305938294019cd4x1f1.jpg

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?

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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\

Learn how to complete the square for the quadratic equation y = x^2 - 6x + 1. This problem includes key algebraic concepts, such as rewriting the equation in vertex form and finding the vertex of the parabola. Suitable for high school students in Grades 9-10.

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