How to Factor Difference of Perfect Squares
The factorization of the difference of two perfect squares is a fundamental concept in algebra. It is a process that involves expressing the difference between two square numbers as a product of two binomials. This technique is not only useful in solving algebraic equations but also in various fields such as geometry, trigonometry, and calculus. In this article, we will discuss the steps and methods to factor the difference of perfect squares.
The general form of a difference of two perfect squares is represented as \(a^2 – b^2\). To factor this expression, we can use the following formula:
\[a^2 – b^2 = (a + b)(a – b)\]
This formula is known as the difference of squares identity. Let’s break down the steps to factor a difference of perfect squares:
1. Identify the two perfect squares: Look for the terms \(a^2\) and \(b^2\) in the given expression. These are the square numbers we will be working with.
2. Apply the difference of squares identity: Use the formula \(a^2 – b^2 = (a + b)(a – b)\) to factor the expression. Replace \(a\) with the first square number and \(b\) with the second square number.
3. Simplify the binomials: If the binomials have any common factors, simplify them. For example, if \(a\) and \(b\) are both even, you can factor out a common factor of 2.
4. Verify the factorization: Multiply the two binomials obtained in step 3 and check if the result matches the original expression. This step is crucial to ensure that the factorization is correct.
Let’s consider a few examples to illustrate the process:
Example 1:
Factor the difference of perfect squares \(16 – 9\).
Solution:
The two perfect squares are \(16\) and \(9\). Applying the difference of squares identity, we have:
\[16 – 9 = (4 + 3)(4 – 3) = 7 \times 1 = 7\]
Example 2:
Factor the difference of perfect squares \(25x^2 – 16\).
Solution:
The two perfect squares are \(25x^2\) and \(16\). Applying the difference of squares identity, we have:
\[25x^2 – 16 = (5x + 4)(5x – 4)\]
In conclusion, the factorization of the difference of perfect squares is a straightforward process that involves identifying the two square numbers, applying the difference of squares identity, simplifying the binomials, and verifying the factorization. By following these steps, you can successfully factor any difference of perfect squares.
