https://file.aiquickdraw.com/mathbot/file/1730561288746dfsf9zbh.jpg

The question is to find the \( x \)-intercepts and the vertex of the parabola given by the equation:

\[
y = x^2 + 6x + 9
\]

### Step 1: Finding the \( x \)-Intercept(s)
To find the \( x \)-intercepts, we set \( y = 0 \):

\[
x^2 + 6x + 9 = 0
\]

This is a quadratic equation, and we can factor it:

\[
(x + 3)(x + 3) = 0
\]

Thus,

\[
x + 3 = 0 \Rightarrow x = -3
\]

So, there is a single \( x \)-intercept at \( x = -3 \). The coordinates of the \( x \)-intercept are \((-3, 0)\).

### Step 2: Finding the Vertex
Since the equation \( y = x^2 + 6x + 9 \) is in standard form \( y = ax^2 + bx + c \), where \( a = 1 \) and \( b = 6 \), we can find the \( x \)-coordinate of the vertex using the formula:

\[
x = -\frac{b}{2a} = -\frac{6}{2 \cdot 1} = -3
\]

To find the \( y \)-coordinate of the vertex, substitute \( x = -3 \) back into the equation:

\[
y = (-3)^2 + 6(-3) + 9 = 9 - 18 + 9 = 0
\]

Thus, the vertex of the parabola is at \((-3, 0)\).

### Summary
- **\( x \)-intercept(s):** \((-3, 0)\)
- **Vertex:** \((-3, 0)\)

The image solution appears to have an error because it lists the \( x \)-intercepts as \(3\) and \(0\) and the vertex as \((3, 0)\). The correct values should be:

- **\( x \)-intercept:** \((-3, 0)\)
- **Vertex:** \((-3, 0)\)

Would you like further details or have any questions?

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#### Related Questions
1. How do you find the \( x \)-intercepts for a quadratic equation in general?
2. What are alternative methods to find the vertex of a quadratic function?
3. Why does the parabola only have one \( x \)-intercept here?
4. How can you rewrite \( y = x^2 + 6x + 9 \) in vertex form?
5. What is the significance of the vertex in a quadratic function?

#### Tip
Remember that the vertex of a parabola in the form \( y = ax^2 + bx + c \) can also indicate the minimum or maximum point, depending on whether the parabola opens upward or downward.

Find the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 6x + 9. If there is more than one x-intercept, separate them with commas.

Explore how to find the x-intercepts and the vertex for the quadratic equation y = x^2 + 6x + 9. This step-by-step guide covers key algebraic techniques including factoring and using the vertex formula, designed for students in Grades 9-10.

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