Problem 82 - Introductory Algebra for College Students

Question

Factor completely. $$x^{2}+\frac{2}{3} x+\frac{1}{9}$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \((x + 1/3)^2\).

Step 1: Identify the Perfect Square Trinomial

The given quadratic expression can be represented in the form \(a^2 + 2ab + b^2\), where \(a = x\), \(b = 1/3\) such that: 1. \(a^2 = x^2\), 2. \(2ab = 2*x*1/3 = 2x/3\), and 3. \(b^2 = (1/3)^2 = 1/9\). This confirms the quadratic is a perfect square trinomial.

Step 2: Factorize the Trinomial

Since the given expression is a perfect square trinomial, it can be factored into \((a+b)^2\). Substituting the values of \(a = x\) and \(b = 1/3\) into the equation, we get \((x + 1/3)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It has the form \(a^2 + 2ab + b^2\).
This type of expression results from squaring a binomial.
The completed square results in

• the first term being \(a^2\)
• the second term being \(2ab\)
• the third term being \(b^2\)

Let's consider these terms in our problem. Here, the perfect square trinomial is \(x^{2} + \frac{2}{3}x + \frac{1}{9}\).
Notice how \(x^2\) is \(a^2\), \(\frac{2}{3}x\) is \(2ab\), and \(\frac{1}{9}\) is \(b^2\).
By identifying \(a = x\) and \(b = \frac{1}{3}\), we confirm this is a perfect square trinomial as it matches the formula pattern.
This understanding helps us factor the trinomial easily as it is already in a recognizable form. The expression simplifies neatly to \((x + \frac{1}{3})^2\).

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations.
It enables the solving of problems and the manipulation of mathematical expressions.
When working with expressions like \(x^{2} + \frac{2}{3}x + \frac{1}{9}\), algebraic manipulation is key.

• Identify patterns: Look for recognizable patterns like the perfect square trinomial.
• Substitute correctly: Use substitution to identify parts of the expression, such as \(a = x\) and \(b = \frac{1}{3}\).
• Factor: Based on the identified pattern, rewrite the expression in its factored form, \((x + \frac{1}{3})^2\).

These steps help break down complex expressions into simpler, more manageable forms.
This fundamental approach in algebra makes problem-solving intuitive and logical.

Quadratic Expression

Quadratic expressions are a crucial part of algebra, characterized by the standard form \( ax^2 + bx + c \).
Quadratics can describe curves such as parabolas and are used in solving equations where variables are squared. The given expression, \(x^{2} + \frac{2}{3}x + \frac{1}{9}\), is an example of a quadratic expression.

• The term \(x^2\) represents the quadratic term
• \(\frac{2}{3}x\) as the linear term
• \(\frac{1}{9}\) as the constant term

Understanding the role of these terms helps identify the structure of the quadratic.
Once the structure is recognized, especially if it's a special form like the perfect square trinomial, factoring into \((x + \frac{1}{3})^2\) becomes straightforward.
This process is essential for simplifying problems and finding solutions in more complex scenarios.