Proof of (a + b)^2 = a^2 + 2ab + b^2
The equation (a + b)^2 = a^2 + 2ab + b^2 is a fundamental algebraic identity that is frequently used in various mathematical operations. This identity states that squaring the sum of two numbers 'a' and 'b' is equivalent to the sum of the squares of 'a' and 'b' plus twice the product of 'a' and 'b'.
Here are two common methods to prove this identity:
1. Expanding the Left Hand Side:
 Start with the lefthand side of the equation: (a + b)^2
 Expand the square: (a + b) * (a + b)
 Apply the distributive property (FOIL): a(a + b) + b(a + b)
 Distribute further: a^2 + ab + ba + b^2
 Combine like terms: a^2 + 2ab + b^2
Therefore, we have shown that (a + b)^2 = a^2 + 2ab + b^2
2. Geometric Proof:
This proof utilizes the concept of areas. Consider a square with sides of length (a + b).

Area of the entire square: (a + b)^2

Divide the square into smaller squares and rectangles:
 A square with side 'a' has area a^2
 A square with side 'b' has area b^2
 Two rectangles with sides 'a' and 'b' each have area ab.

Adding the areas of the smaller shapes: a^2 + b^2 + ab + ab = a^2 + 2ab + b^2
This shows that the area of the entire square (a + b)^2 is equal to the sum of the areas of the smaller squares and rectangles (a^2 + 2ab + b^2).
Both methods demonstrate that (a + b)^2 = a^2 + 2ab + b^2 is a true and useful identity in mathematics. It allows us to simplify and manipulate algebraic expressions, making calculations easier and more efficient.