Special Identities

Identity 1A: Square of a Sum

When we multiply a number by itself, we get a square. If that number is a sum $(a+b)$, we are finding the area of a square with side length $(a+b)$.

$(a + b)^2 = a^2 + 2ab + b^2$

Geometric Proof

Imagine a large square with side $a+b$. We can split it into 4 sections:

1. A square of area $a^2$.
2. A square of area $b^2$.
3. Two identical rectangles of area $a \times b$.

Identity 1B: Square of a Difference

$(a - b)^2 = a^2 - 2ab + b^2$

Derivation: We can derive this from Identity 1A by replacing $b$ with $(-b)$: $(a + (-b))^2 = a^2 + 2a(-b) + (-b)^2$ $= a^2 - 2ab + b^2$

Identity 1C: Difference of Squares

This is a very useful pattern for fast calculation. $(a + b)(a - b) = a^2 - b^2$

Why does this work? Let’s expand it using the distributive property: $(a + b)(a - b) = a(a - b) + b(a - b)$ $= a^2 - ab + ab - b^2$ $= a^2 - b^2$ (The middle terms $-ab$ and $+ab$ cancel out!)