The quadratic trinomial you provided is:

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

### Factoring the Trinomial:
Let's factor this trinomial. To factor \( x^2 + 6x + 9 \), we look for two numbers that multiply to \( 9 \) (the constant term) and add to \( 6 \) (the coefficient of the linear term).

These two numbers are \( 3 \) and \( 3 \), because:

\[
3 \times 3 = 9 \quad \text{and} \quad 3 + 3 = 6
\]

Thus, the trinomial can be factored as:

\[
x^2 + 6x + 9 = (x + 3)(x + 3)
\]

This can also be written as:

\[
(x + 3)^2
\]

### Interpretation:
- The quadratic trinomial \( x^2 + 6x + 9 \) is a **perfect square trinomial**.
- It factors into \( (x + 3)^2 \), meaning it has a repeated root.
- The root is \( x = -3 \), because \( (x + 3) = 0 \) leads to \( x = -3 \).

### Discriminant:
The discriminant \( \Delta \) for the trinomial \( ax^2 + bx + c \) is calculated as:

\[
\Delta = b^2 - 4ac
\]

For \( x^2 + 6x + 9 \), we have \( a = 1 \), \( b = 6 \), and \( c = 9 \):

\[
\Delta = 6^2 - 4(1)(9) = 36 - 36 = 0
\]

Since the discriminant is \( 0 \), the trinomial has exactly one real root (a repeated root), confirming our factorization.

### Graph:
The graph of the quadratic function \( y = x^2 + 6x + 9 \) is a parabola that opens upwards (since the leading coefficient \( a = 1 \) is positive) and has its vertex at the point \( (-3, 0) \). The vertex represents the minimum point of the parabola.

Would you like to explore any particular part of this further, or do you have any other questions?

### Related Questions:
1. How do you recognize a perfect square trinomial?
2. What is the process to factor other quadratic trinomials?
3. How can you derive the vertex of a quadratic function from its standard form?
4. What is the significance of the discriminant being zero?
5. How would the factorization change if the constant term were different?
6. What are the steps to graph a quadratic function like \( x^2 + 6x + 9 \)?
7. How can completing the square be used to derive the factorization?
8. How would the graph change if the coefficient of \( x^2 \) were negative?

### Tip:
When you factor a quadratic trinomial and get a repeated root (like \( (x + 3)^2 \)), it indicates that the graph of the function touches the x-axis at exactly one point, known as the vertex.

Learn how to factor the quadratic trinomial x^2 + 6x + 9 using the perfect square trinomial method. Explore the discriminant and vertex of the related parabola. Suitable for high school students studying quadratic equations.

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