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

Jasonbradley

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Jun 17, 2024

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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.