The expression \(x^2 + 11x + 18\) is a general quadratic trinomial of the form \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 11\), and \(c = 18\).

### 1. Factoring the Quadratic Trinomial

To factor the trinomial \(x^2 + 11x + 18\), we need to find two numbers that multiply to \(c = 18\) and add up to \(b = 11\). 

Let's break it down:

1. The factors of 18 are: \(1 \times 18\), \(2 \times 9\), and \(3 \times 6\).
2. Out of these pairs, the pair \(2\) and \(9\) add up to 11.

So, we can factor the quadratic as:
\[
x^2 + 11x + 18 = (x + 2)(x + 9)
\]

### 2. Roots of the Quadratic Equation

If you were to set the quadratic trinomial equal to zero:
\[
x^2 + 11x + 18 = 0
\]
You can solve for \(x\) by setting each factor to zero:
\[
x + 2 = 0 \quad \text{or} \quad x + 9 = 0
\]
Thus, the roots are:
\[
x = -2 \quad \text{and} \quad x = -9
\]

### 3. Vertex of the Parabola

The quadratic function \(y = x^2 + 11x + 18\) represents a parabola. The vertex of the parabola can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{11}{2(1)} = -\frac{11}{2} = -5.5
\]
Substitute \(x = -5.5\) back into the equation to find the y-coordinate:
\[
y = (-5.5)^2 + 11(-5.5) + 18 = 30.25 - 60.5 + 18 = -12.25
\]
So, the vertex is at \((-5.5, -12.25)\).

### 4. Axis of Symmetry

The axis of symmetry for the parabola is the vertical line that passes through the vertex:
\[
x = -5.5
\]

### 5. Direction of the Parabola

Since the coefficient of \(x^2\) (i.e., \(a = 1\)) is positive, the parabola opens upwards.

Would you like to explore any of these topics further, or have any questions? Here are some related questions:

1. How do you complete the square for the quadratic \(x^2 + 11x + 18\)?
2. What is the discriminant of the quadratic equation, and how does it determine the number of roots?
3. How would the graph change if the coefficient \(a\) were negative?
4. How do you derive the quadratic formula?
5. What is the impact of changing the constant term \(c\) in the quadratic equation?

**Tip:** Factoring is often the quickest method to solve simple quadratic equations, but the quadratic formula is universally applicable.

Explore how to factor and solve the quadratic trinomial x^2 + 11x + 18 step-by-step. Learn about factoring techniques, finding the vertex and axis of symmetry of the parabola, and determining the direction of the parabola. Suitable for high school students studying algebra and quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation