The difference of squares is a specific factoring pattern that allows you to factor expressions of the form a²-b² into (a+b)(a-b).
Understanding Difference of Squares
The core concept of factoring using the difference of squares lies in recognizing the pattern where an expression is the subtraction of two perfect squares. This pattern follows the formula:
a² - b² = (a + b)(a - b)
This formula is derived from the expansion of (a+b)(a-b), which results in a² - b².
Steps to Factor Using Difference of Squares
- Identify Perfect Squares: Determine if the terms in the expression are perfect squares. A perfect square is a number or variable that can be expressed as the square of another number or variable (e.g., x², 9, 16, 25y²).
- Set Up Parentheses: Once you have identified the terms, set up two sets of parentheses like this: ( )( ).
- Find the Square Roots: Calculate the square root of each perfect square identified in step 1. Let these square roots be 'a' and 'b'.
- Insert into Parentheses: Place 'a' and 'b' into the parentheses according to the difference of squares pattern: (a+b)(a-b).
- Verify: Expand the factored expression (a+b)(a-b) to verify that it matches the original expression (a²-b²).
Examples of Factoring Using Difference of Squares
Here are some practical examples to illustrate the concept:
- Example 1: Factoring x² - 25
- x² is a perfect square (x * x).
- 25 is a perfect square (5 * 5).
- Using the difference of squares formula, x²-25 becomes (x+5)(x-5).
- Example 2: Factoring 4y² - 9
- 4y² is a perfect square (2y * 2y).
- 9 is a perfect square (3 * 3).
- Using the difference of squares formula, 4y²-9 becomes (2y+3)(2y-3).
- Example 3: Factoring 16x² - 81y²
- 16x² is a perfect square (4x * 4x).
- 81y² is a perfect square (9y * 9y).
- Using the difference of squares formula, 16x² - 81y² becomes (4x + 9y)(4x - 9y).
- Example 4: Factoring 1 - a²
- 1 is a perfect square (1 * 1).
- a² is a perfect square (a * a).
- Using the difference of squares formula, 1 - a² becomes (1 + a)(1 - a).
Why Does This Work?
The reason this method works is based on the distributive property of multiplication:
(a+b)(a-b) = a(a-b) + b(a-b) = a² - ab + ba - b² = a² - b²
As you can see, the middle terms (-ab and +ba) cancel each other out, leaving only a² - b².
Tips for Recognizing the Difference of Squares
- Two Terms: The expression must have exactly two terms.
- Subtraction Sign: There must be a subtraction sign between the terms.
- Perfect Squares: Both terms must be perfect squares.
By remembering these points, you can easily recognize expressions that can be factored using the difference of squares method.
