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The Expansion of (x+y)²: A Mathematical Formula
In algebra, one of the most fundamental formulas is the expansion of (x+y)², which is equal to x² + 2xy + y². This formula is widely used in various mathematical operations, including squares, quadratic equations, and geometric calculations.
The Formula: (x+y)² = x² + 2xy + y²
The formula (x+y)² = x² + 2xy + y² is a mathematical identity that expresses the square of a binomial expression (x+y) as the sum of three terms: x², 2xy, and y². This formula can be proven through geometric and algebraic methods.
Geometric Proof
One way to prove the formula is by using geometric representation. Consider a square with side length (x+y). The area of the square can be calculated as (x+y)².
<h3 align="center"> (x+y)² </h3>
The square can be divided into four rectangles, as shown below:
<h3 align="center"> x² | xy
xy | y² </h3>
The area of the square is equal to the sum of the areas of the four rectangles:
(x+y)² = x² + 2xy + y²
Algebraic Proof
Another way to prove the formula is by using algebraic manipulation. Start with the binomial expression (x+y)² and expand it using the distributive property:
(x+y)² = (x+y)(x+y)
= x(x+y) + y(x+y)
= x² + xy + yx + y²
= x² + 2xy + y²
Applications of the Formula
The formula (x+y)² = x² + 2xy + y² has numerous applications in various branches of mathematics, including:
Quadratic Equations
The formula is used to solve quadratic equations of the form ax² + bx + c = 0, where a, b, and c are constants.
Geometric Calculations
The formula is used to calculate the area and perimeter of various geometric shapes, such as squares, rectangles, and triangles.
Algebraic Manipulations
The formula is used to simplify algebraic expressions and to factorize quadratic expressions.
Conclusion
In conclusion, the formula (x+y)² = x² + 2xy + y² is a fundamental mathematical identity that has numerous applications in algebra, geometry, and other branches of mathematics. By understanding and applying this formula, students and mathematicians can simplify complex mathematical operations and solve problems more efficiently.
