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Understanding the (x + y)² Formula
The formula (x + y)² represents the square of the sum of two variables, x and y. Expanding this formula is a fundamental concept in algebra and has wide applications in various mathematical fields.
Expanding the Formula
To expand (x + y)², we use the distributive property (also known as FOIL method). Here's how it works:
1. Square the first term: x * x = x²
2. Multiply the first and second terms and double the result: 2 * (x * y) = 2xy
3. Square the second term: y * y = y²
4. Combine all the terms: x² + 2xy + y²
Therefore, (x + y)² = x² + 2xy + y²
Applications
The (x + y)² formula has numerous applications in mathematics, including:
- Factoring quadratic equations: This formula is crucial for factoring quadratic expressions in the form ax² + bx + c.
- Simplifying algebraic expressions: It helps simplify expressions involving squares of sums or differences.
- Solving equations: It can be used to solve equations involving squares of sums or differences.
- Geometry: The formula can be applied to find the area of squares or rectangles.
Example
Problem: Expand the expression (2a + 3b)²
Solution: Using the (x + y)² formula, we can substitute:
- x = 2a
- y = 3b
Therefore, (2a + 3b)² = (2a)² + 2(2a)(3b) + (3b)² = 4a² + 12ab + 9b²
Conclusion
Understanding the (x + y)² formula and its expansion is essential for solving various mathematical problems. By mastering this fundamental concept, you can simplify algebraic expressions, factor quadratic equations, and tackle more complex mathematical challenges.