(x+y)^2 = X^2 + Y^2

2 min read Jun 17, 2024
(x+y)^2 = X^2 + Y^2

The Common Misconception: (x + y)² ≠ x² + y²

One of the most common mistakes in algebra is assuming that (x + y)² is equal to x² + y². While it may seem intuitive, this is simply not true.

Understanding the Difference

The correct expansion of (x + y)² is x² + 2xy + y². This is derived from the distributive property of multiplication.

Let's break down why:

  • (x + y)² means (x + y) * (x + y).
  • Expanding this using the distributive property, we get:
    • x(x + y) + y(x + y)
    • x² + xy + xy + y²
    • x² + 2xy + y²

Visualizing the Difference

Imagine a square with sides of length x + y.

  • The area of the square is (x + y)².
  • This square can be divided into four smaller rectangles:
    • One with sides x by x (area x²)
    • One with sides y by y (area y²)
    • Two rectangles with sides x by y (area xy each).

This visualization clearly shows that the area of the entire square is not simply x² + y², but includes the additional term 2xy which represents the areas of the two smaller rectangles.

Importance of Correct Understanding

Failing to correctly expand (x + y)² can lead to significant errors in various mathematical problems and applications, including:

  • Solving equations
  • Finding derivatives and integrals
  • Analyzing functions
  • Working with geometric formulas

Always remember: (x + y)² is not equal to x² + y². The correct expansion is x² + 2xy + y².

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