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The Difference of Squares: (x-y)(x+y)
In mathematics, the expression (x-y)(x+y) is a common and important pattern known as the difference of squares. Understanding this pattern can significantly simplify algebraic expressions and calculations.
Expanding the Expression
To see why (x-y)(x+y) is called the difference of squares, let's expand the expression:
- Step 1: Apply the distributive property.
- (x-y)(x+y) = x(x+y) - y(x+y)
- Step 2: Distribute further.
- x(x+y) - y(x+y) = x² + xy - xy - y²
- Step 3: Simplify by combining like terms.
- x² + xy - xy - y² = x² - y²
The Result: x² - y²
As you can see, expanding (x-y)(x+y) results in x² - y², which is the difference of two squares: x² and y². This is why the expression is called the difference of squares.
Applications and Significance
Understanding the difference of squares pattern is crucial in algebra for several reasons:
- Factoring: It allows you to easily factor expressions in the form x² - y² into (x-y)(x+y). This is helpful for solving equations and simplifying expressions.
- Simplifying expressions: The pattern can be used to quickly simplify more complex expressions involving the difference of squares.
- Solving equations: The difference of squares pattern can be used to solve quadratic equations of the form ax² - c = 0.
Examples
Here are some examples of how the difference of squares pattern can be applied:
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Factoring: Factor the expression x² - 9.
- Notice that 9 is a perfect square (3²).
- Therefore, x² - 9 can be factored as (x - 3)(x + 3).
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Simplifying expressions: Simplify the expression (2x - 3y)(2x + 3y).
- Using the difference of squares pattern, we get: (2x)² - (3y)² = 4x² - 9y²
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Solving equations: Solve the equation x² - 16 = 0.
- Factor the equation: (x - 4)(x + 4) = 0
- Set each factor to zero and solve:
- x - 4 = 0 => x = 4
- x + 4 = 0 => x = -4
Conclusion
The difference of squares pattern is a fundamental concept in algebra with numerous applications in simplifying expressions, factoring, and solving equations. By understanding and applying this pattern, you can significantly improve your understanding and proficiency in algebra.