1. The Basics: Squares and Differences

These are the most common identities used to expand or factorize quadratic expressions.

• Difference of Squares: $(a + b)(a – b) = a^2 – b^2$

• Square of a Sum: $(a + b)^2 = a^2 + b^2 + 2ab$

• Square of a Difference: $(a – b)^2 = a^2 + b^2 – 2ab$

2. Relationship Identities

Sometimes you need to convert between a sum and a difference. These formulas show how $(a+b)^2$ and $(a-b)^2$ relate to one another.

• $(a + b)^2 + (a – b)^2 = 2(a^2 + b^2)$

• $(a + b)^2 – (a – b)^2 = 4ab$

• $(a + b)^2 = (a – b)^2 + 4ab$

• $(a – b)^2 = (a + b)^2 – 4ab$

• Finding the sum of squares: $a^2 + b^2 = (a + b)^2 – 2ab$

  OR

  $a^2 + b^2 = (a – b)^2 + 2ab$

3. Expanding Three Variables

When dealing with three terms $(a, b, c)$, the signs of the product terms depend on the signs within the parentheses.

• $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

• $(a + b – c)^2 = a^2 + b^2 + c^2 + 2ab – 2bc – 2ca$

• $(a – b + c)^2 = a^2 + b^2 + c^2 – 2ab – 2bc + 2ca$

• $(a – b – c)^2 = a^2 + b^2 + c^2 – 2ab + 2bc – 2ca$

4. Cubic Identities

Cubic identities are vital for higher-level algebra and calculus.

• Cube of a Sum: $(a + b)^3 = a^3 + b^3 + 3ab(a + b) = a^3 + 3a^2b + 3ab^2 + b^3$

• Cube of a Difference: $(a – b)^3 = a^3 – b^3 – 3ab(a – b) = a^3 – 3a^2b + 3ab^2 – b^3$

• Sum of Two Cubes: $a^3 + b^3 = (a + b)^3 – 3ab(a + b) = (a + b)(a^2 – ab + b^2)$

• Difference of Two Cubes: $a^3 – b^3 = (a – b)^3 + 3ab(a – b) = (a – b)(a^2 + ab + b^2)$

5. The Advanced Triple Cube Identity

This is a specialized identity often used in advanced factorization problems.

• $a^3 + b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 – ab – bc – ca)$

• Special Condition: If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$.