(a+b)^2 = A^2 + B^2 Proof

4 min read Jun 16, 2024

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 cross-multiplication 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 cross-multiplication terms when expanding squared expressions.