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Factoring Difference of Squares: A Step-by-Step Guide
In algebra, factoring is a crucial skill that helps simplify expressions and solve equations. One common factoring pattern is the difference of squares, which allows us to break down expressions that have a specific structure. This guide will provide a step-by-step explanation of how to factor difference of squares expressions, along with examples to illustrate the process.
Understanding the Difference of Squares Pattern
The difference of squares pattern is based on the following algebraic identity:
a² – b² = (a + b)(a – b)
This identity states that the difference of two perfect squares (a² and b²) can be factored into the product of the sum and difference of their square roots (a + b and a – b).
Steps to Factor Difference of Squares
Follow these steps to factor expressions that fit the difference of squares pattern:
- Identify the perfect squares: Look for terms that are perfect squares. A perfect square is a number or variable that can be obtained by squaring another number or variable. For example, 4, 9, 16, x², y², and 25a² are all perfect squares.
- Check for subtraction: Ensure that the expression involves subtraction between the two perfect squares.
- Factor using the formula: Apply the difference of squares formula: a² – b² = (a + b)(a – b). Find the square roots of the perfect squares and plug them into the formula.
Examples
Let’s illustrate the process with some examples:
Example 1:
Factor the expression x² – 9.
1. Identify the perfect squares: x² is the square of x, and 9 is the square of 3.
2. Check for subtraction: The expression involves subtraction.
3. Apply the formula: x² – 9 = (x + 3)(x – 3)
Example 2:
Factor the expression 4y² – 25.
1. Identify the perfect squares: 4y² is the square of 2y, and 25 is the square of 5.
2. Check for subtraction: The expression involves subtraction.
3. Apply the formula: 4y² – 25 = (2y + 5)(2y – 5)
Additional Resources
For further practice and exploration, you can refer to these resources:
Conclusion
Factoring difference of squares is a straightforward technique that can be applied to simplify expressions and solve equations. By understanding the pattern and following the steps outlined in this guide, you can confidently factor expressions that fit this pattern. Remember to practice regularly and consult additional resources for further learning.