The given equation is:

\[
x^2 - 6x - y + 11 = 0
\]

This is a quadratic equation in terms of \(x\) and \(y\). To express this in a form that can be easily analyzed (like the equation of a parabola), we can rearrange it to solve for \(y\). Here's how to proceed:

### Step 1: Rearrange the equation
We start by isolating \(y\) on one side:
\[
y = x^2 - 6x + 11
\]

### Step 2: Complete the square
To better understand the structure of the equation, let's complete the square for the \(x^2 - 6x\) part.

1. Take half of the coefficient of \(x\) (which is \(-6\)), and square it:
   \[
   \left( \frac{-6}{2} \right)^2 = 9
   \]

2. Now, add and subtract 9 inside the equation to complete the square:
   \[
   y = (x^2 - 6x + 9) + 11 - 9
   \]
   Simplifying this gives:
   \[
   y = (x - 3)^2 + 2
   \]

### Final form of the equation
\[
y = (x - 3)^2 + 2
\]

This is the equation of a **parabola** that opens upwards. The vertex of the parabola is at \((3, 2)\).

### Key points:
- The vertex form of the equation is \(y = (x - 3)^2 + 2\), with the vertex at \((3, 2)\).
- The parabola opens upwards since the coefficient of the squared term is positive.
- The minimum value of \(y\) is 2, which occurs when \(x = 3\).

Would you like more details or have any questions?

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Here are five related questions to further your understanding:
1. How do you find the axis of symmetry of a parabola?
2. What is the significance of completing the square in analyzing quadratic equations?
3. How can you determine whether a parabola opens upwards or downwards?
4. What is the role of the vertex in the graph of a quadratic equation?
5. How do you find the x-intercepts and y-intercepts of a quadratic equation?

**Tip:** Completing the square is a useful technique to transform a quadratic equation into vertex form, making it easier to analyze its geometric properties.

Explore how to solve the quadratic equation X²-6x-y+11=0 by completing the square. This solution transforms the equation into vertex form, revealing the structure of the parabola with its vertex at (3, 2). Perfect for Grades 9-10 students learning about parabolas and completing the square.

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