Problem 61 - Elementary Algebra

Question

For the following problems, find the products. $$ \left(a+\frac{2}{9}\right)\left(a-\frac{2}{9}\right) $$

Step-by-Step Solution

Answer

Answer: The product of the binomials is $$a^2 - \frac{4}{81}$$.

Step 1: Identify the problem type and formula

This is an example of the difference of squares problem, and we will use the formula \((a^2 - k^2)\) to find the product.

Step 2: Apply the difference of squares formula

Use the given binomials and apply the formula: \((a + \frac{2}{9})(a - \frac{2}{9}) = a^2 - \left(\frac{2}{9}\right)^2\)

Step 3: Calculate the square of the fraction

Square the fraction: \(\left(\frac{2}{9}\right)^2 = \frac{4}{81}\)

Step 4: Find the final solution

Replace the square of the fraction in the equation: \(a^2 - \left(\frac{2}{9}\right)^2 = a^2 - \frac{4}{81}\) So, the product is: $$ \left(a+\frac{2}{9}\right)\left(a-\frac{2}{9}\right) = a^2 - \frac{4}{81} $$

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics that allow us to express relationships using numbers and symbols. In general, they consist of variables, constants, and operators (such as addition, subtraction, etc.). These expressions can be manipulated following certain rules to simplify or solve equations.

For example, in the exercise, we work with the expression \((a + \frac{2}{9})(a - \frac{2}{9})\). Here, \(a\) is a variable, and \(\frac{2}{9}\) is a constant. Understanding how to work with these components is key to manipulating expressions and finding solutions in algebra.

A good grasp of algebraic expressions helps in breaking down complex problems and analyzing them step-by-step. This problem particularly demonstrates how they can be combined and simplified using identities.

Factoring Binomials

Factoring binomials involves breaking down a polynomial into simpler components (or factors) that, when multiplied together, give you the original polynomial. This is a powerful technique in algebra because it makes equations easier to solve or simplifies expressions.

In our case, we deal with the binomial \((a + \frac{2}{9})(a - \frac{2}{9})\). When these binomials are factored, they simplify into the difference of squares identity \(a^2 - \left(\frac{2}{9}\right)^2\).

The idea is to recognize patterns like the difference of squares, which here simplify to \(a^2 - \frac{4}{81}\). This method of factoring is useful as it applies to many algebraic identities and lets you solve or simplify equations efficiently. Thus, being familiar with common binomial products helps you tackle a variety of algebraic problems.

Quadratic Identities

Quadratic identities provide formulas that help us simplify expressions and solve equations involving squared terms. One important identity is the difference of squares, which states that \((x + y)(x - y) = x^2 - y^2\).

In the given exercise, this identity simplifies the product of the binomials \((a + \frac{2}{9})(a - \frac{2}{9})\) into \(a^2 - \left(\frac{2}{9}\right)^2\). This shows that recognizing and applying quadratic identities can greatly simplify otherwise complex algebraic problems.

Understanding these identities is crucial for students as they progress through algebra, providing them with tools to solve equations more efficiently. Whether solving for variables or simplifying algebraic expressions, quadratic identities are a key component in a mathematician's toolkit.