The Misconception: (a  b)^2 = a^2  b^2
The equation (a  b)^2 = a^2  b^2 is a common misconception in algebra. While it may seem intuitive at first glance, it is incorrect.
Why It's Wrong
The correct expansion of (a  b)^2 is (a  b)^2 = a^2  2ab + b^2.
Let's break it down:

(a  b)^2 represents the square of the binomial (a  b). This means we are multiplying the binomial by itself: (a  b)^2 = (a  b)(a  b)

To expand this, we use the distributive property (also known as FOIL  First, Outer, Inner, Last): (a  b)(a  b) = a(a  b)  b(a  b)

Applying the distributive property again: a(a  b)  b(a  b) = a^2  ab  ba + b^2

Combining like terms: a^2  ab  ba + b^2 = a^2  2ab + b^2
The Importance of Correct Expansion
Understanding the correct expansion of (a  b)^2 is crucial for various reasons:
 Accuracy in calculations: Using the incorrect formula will lead to inaccurate results.
 Solving equations: Many mathematical problems involve squaring binomials. Using the correct expansion is essential for finding the correct solutions.
 Understanding algebraic concepts: Mastering the expansion of binomials is a fundamental concept in algebra and paves the way for understanding more complex algebraic expressions.
Conclusion
Remember, (a  b)^2 = a^2  2ab + b^2, not a^2  b^2. Pay attention to the middle term (2ab) and avoid falling into the trap of this common misconception.