The Misconception: (a + b)^2 = a^2 + b^2
It is a common misconception that squaring a sum of two terms is the same as squaring each term individually. In other words, many believe that (a + b)^2 = a^2 + b^2. However, this is incorrect. This article will demonstrate why this equation is false and provide the correct expansion of (a + b)^2.
The Correct Expansion:
The correct expansion of (a + b)^2 is:
(a + b)^2 = a^2 + 2ab + b^2
Let's break down why this is the case:
Understanding the Square:
Squaring a term means multiplying it by itself. So, (a + b)^2 is the same as (a + b) * (a + b).
To expand this, we can use the distributive property:

Step 1: Multiply the first term in the first bracket by each term in the second bracket:
 a * a = a^2
 a * b = ab

Step 2: Multiply the second term in the first bracket by each term in the second bracket:
 b * a = ba (which is the same as ab)
 b * b = b^2
Now we have: a^2 + ab + ba + b^2
Finally, combining the like terms, we get:
(a + b)^2 = a^2 + 2ab + b^2
The Importance of the Middle Term:
The crucial difference between the incorrect equation and the correct one is the middle term, 2ab. This term arises from the crossmultiplication of 'a' and 'b' during the expansion process. It is essential to include this term to obtain the accurate result.
Visual Representation:
A visual representation can help understand this concept. Imagine a square with side length (a + b). The area of this square is (a + b)^2.
We can divide this square into four smaller squares and two rectangles:
 Square 1: Side length 'a', area 'a^2'
 Square 2: Side length 'b', area 'b^2'
 Rectangle 1: Length 'a', width 'b', area 'ab'
 Rectangle 2: Length 'b', width 'a', area 'ab'
The total area of the larger square is the sum of the areas of all the smaller squares and rectangles: a^2 + ab + ab + b^2 = a^2 + 2ab + b^2
Conclusion:
It is important to remember that (a + b)^2 is not equal to a^2 + b^2. The correct expansion is a^2 + 2ab + b^2. Always remember to consider the crossmultiplication terms when expanding squared expressions.