Problem 21 - Elementary and Intermediate Algebra

Question

Factor. $$ x^{2}+6 x+9 $$

Step-by-Step Solution

Answer

The expression factors to \((x + 3)^2\).

Step 1: Recognize the Quadratic Form

The expression is a quadratic trinomial of the form \( ax^2 + bx + c \). In this case, \( a = 1 \), \( b = 6 \), and \( c = 9 \). Our task is to factor this expression.

Step 2: Check for Perfect Square Trinomial

A perfect square trinomial is of the form \( (x + d)^2 \) and expands to \( x^2 + 2dx + d^2 \). Compare this with the given expression \( x^2 + 6x + 9 \) to check if it fits this pattern.

Step 3: Identify \( d \)

For a perfect square trinomial \( x^2 + 6x + 9 \), equate \( 2d = 6 \) to find \( d \). Solving \( 2d = 6 \) gives \( d = 3 \). Also, check if \( d^2 = 9 \). Since \( 3^2 = 9 \), the expression fits the pattern of a perfect square trinomial.

Step 4: Write the Factored Form

Since \( x^2 + 6x + 9 \) is a perfect square trinomial, it can be factored as \( (x + 3)^2 \). Therefore, the factored form of the expression is \( (x + 3)(x + 3) \).

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are a special type of quadratic trinomial. They have the form \((x + d)^2\), which when expanded, results in \(x^2 + 2dx + d^2\). But what makes them "perfect squares"? It's when both the middle term and the constant term follow a particular pattern.

To recognize a perfect square trinomial, look for these characteristics:

• The coefficient of the linear term, \(b\), should be twice the product of \(d\), which means \(b = 2d\).
• The constant term \(c\) should be the square of \(d\), which means \(c = d^2\).

In practice, for the problem \(x^2 + 6x + 9\):

- We observe \(2d = 6\) which implies \(d = 3\).
- Additionally, \(3^2 = 9\), satisfying the condition \(d^2 = c\).

These conditions are met, confirming it's a perfect square trinomial. This means it can be factored into \((x + 3)^2\). Becoming familiar with these patterns makes factoring quicker and neater.

Quadratic Trinomials

Quadratic trinomials are expressions consisting of three terms and are in the form \(ax^2 + bx + c\). Here, they are called "quadratic" because the highest power of \(x\) is 2. This type of expression is very common in algebra and understanding its components will greatly help in factoring them.

When dealing with quadratic trinomials:

• Identify the coefficients \(a\), \(b\), and \(c\).
• Check patterns or use methods like factoring by grouping to simplify.
• Consider whether it can be a perfect square trinomial if conditions permit.

Specifically in our exercise, we have \(x^2 + 6x + 9\) where \(a = 1\), \(b = 6\), and \(c = 9\). The method involved checking if it was a perfect square trinomial. Once identified, this simplification becomes evident, reducing it further to its squared form.

Factored Form

The factored form of a quadratic expression reveals the product structure of the polynomial and simplifies its components. In essence, factorization is the process of breaking down a complex expression into simpler, multiplicative parts that, when expanded, return the original expression.

In the context of perfect square trinomials, the factored form is particularly simple. For instance, the quadratic \(x^2 + 6x + 9\) is a perfect square trinomial. Here, once it's identified, it can be factored directly to \((x + 3)^2\). Why is this useful? Because:

• It reveals roots easily; here the root is \(-3\).
• The factored form can simplify solving equations that use this expression.
• It provides insights into the geometry of the function when plotted.

Recognizing and converting quadratic expressions into their factored forms is an essential skill in algebra, making equations easier to handle and understand.