(x%2Ba)-formula.jpeg "(x-a)(x+a) Formula")
The Difference of Squares Formula: (x-a)(x+a)
The difference of squares formula is a fundamental algebraic identity that allows us to quickly factor or expand certain expressions. It states:
** (x-a)(x+a) = x² - a² **
This formula is incredibly useful in various algebraic manipulations and problem-solving scenarios. Let's break down its meaning and applications.
Understanding the Formula
The formula tells us that the product of two binomials, where one is the sum of two terms (x+a) and the other is the difference of the same two terms (x-a), simplifies to the difference of the squares of those terms (x² - a²).
Here's how it works:
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FOIL method: We can expand the left side of the equation using the FOIL method (First, Outer, Inner, Last):
(x-a)(x+a) = xx + xa - ax - aa
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Simplifying: The middle terms cancel out:
x² + ax - ax - a² = x² - a²
This demonstrates why the formula is called the difference of squares.
Applications of the Difference of Squares Formula
The difference of squares formula has various applications in algebra and beyond:
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Factoring expressions: You can quickly factor any expression that fits the pattern x² - a². For example, x² - 9 can be factored as (x-3)(x+3).
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Simplifying expressions: By recognizing the pattern, you can simplify complex expressions involving differences of squares.
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Solving equations: The formula can be used to solve quadratic equations where the expression is in the form x² - a² = 0.
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Calculus and other advanced math: The formula is applied in calculus and other advanced math concepts like derivative and integration.
Examples
Here are a few examples to illustrate how the difference of squares formula is used:
1. Factoring:
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Factor x² - 25: Here, x² = x² and 25 = 5², so the expression becomes (x - 5)(x + 5).
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Factor 4y² - 9: Here, 4y² = (2y)² and 9 = 3², so the expression becomes (2y - 3)(2y + 3).
2. Simplifying:
- Simplify (3x - 2y)(3x + 2y): This directly applies the formula, resulting in (3x)² - (2y)² = 9x² - 4y².
3. Solving equations:
- Solve x² - 16 = 0: Factoring using the formula, we get (x - 4)(x + 4) = 0. Therefore, x = 4 or x = -4.
Conclusion
The difference of squares formula is a powerful tool for simplifying and manipulating algebraic expressions. By understanding its structure and applications, you can save time and effort while solving various mathematical problems.