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Factoring the Difference of Squares: (x² - y²)
The expression (x² - y²) is a classic example of a difference of squares. This pattern appears frequently in algebra and can be factored using a simple and elegant formula.
Understanding the Pattern
The difference of squares pattern arises when we have two perfect squares being subtracted from each other. In the expression (x² - y²), we have:
- x²: The square of the variable x
- y²: The square of the variable y
Factoring the Difference of Squares
The formula for factoring the difference of squares is:
(a² - b²) = (a + b)(a - b)
To apply this to (x² - y²), we simply substitute:
- a = x
- b = y
Therefore, the factored form of (x² - y²) is:
(x² - y²) = (x + y)(x - y)
Example
Let's factor the expression (9x² - 4y²):
- Identify the perfect squares: We have 9x² (which is (3x)²) and 4y² (which is (2y)²)
- Apply the formula: (9x² - 4y²) = (3x + 2y)(3x - 2y)
Why Does This Work?
The formula for factoring the difference of squares works because of the distributive property. When you expand (x + y)(x - y), you get:
- x(x - y) + y(x - y)
- x² - xy + xy - y²
- x² - y²
The middle terms (-xy + xy) cancel out, leaving us with the original expression (x² - y²).
Applications
The difference of squares pattern is used extensively in algebra and other areas of mathematics. Some common applications include:
- Solving quadratic equations: By factoring a quadratic equation into the difference of squares, we can easily find its roots.
- Simplifying expressions: Factoring the difference of squares can simplify complex algebraic expressions.
- Solving problems in geometry: The difference of squares pattern can be used to find the area or volume of certain geometric shapes.
Understanding the difference of squares pattern is a crucial step in developing strong algebraic skills. By mastering this pattern, you'll be better equipped to tackle more complex mathematical problems.