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The problem in the image is asking to "Complete the square and find the minimum or maximum value of the quadratic function \( y = x^2 + 16x + 65 \)."

Let's go through the steps to solve this by completing the square.

### Step 1: Start with the given quadratic equation
\[
y = x^2 + 16x + 65
\]

### Step 2: Complete the square
First, we focus on the terms involving \( x \), i.e., \( x^2 + 16x \). To complete the square, take half of the coefficient of \( x \) (which is 16), square it, and add it inside the equation. Then subtract the same value to maintain equality.

- Half of 16 is 8, and \( 8^2 = 64 \).

So, rewrite the equation as:
\[
y = (x^2 + 16x + 64) + 65 - 64
\]
This simplifies to:
\[
y = (x + 8)^2 + 1
\]

### Step 3: Identify the vertex form
Now the equation is in vertex form:
\[
y = (x + 8)^2 + 1
\]
From here, we can see that the vertex of the parabola is \( (-8, 1) \).

### Step 4: Maximum or Minimum
Since the coefficient of \( (x + 8)^2 \) is positive, the parabola opens upwards, meaning it has a **minimum** value at the vertex. The minimum value of \( y \) is 1, and it occurs at \( x = -8 \).

### Answer:
- Minimum value: 1
- It occurs at \( x = -8 \).

It looks like you might have selected "maximum" in the dropdown by mistake in the answer form. The correct option should be "minimum."

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Do you have any questions or would you like further details?

Here are 5 related questions for practice:
1. How do you complete the square for \( y = x^2 - 10x + 22 \)?
2. What is the vertex of the quadratic equation \( y = 2x^2 - 12x + 7 \)?
3. How can you tell if a parabola opens upwards or downwards?
4. What are the benefits of expressing a quadratic function in vertex form?
5. How do you find the axis of symmetry of a quadratic function?

**Tip:** When completing the square, always ensure you balance the equation by both adding and subtracting the same value.

Complete the square and find the minimum or maximum value of the quadratic function y = x^2 + 16x + 65.

Learn how to complete the square for the quadratic function y = x^2 + 16x + 65. This tutorial walks through the process to find the minimum value by converting the equation into vertex form, ideal for students in Grades 9-11.

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