Algebraic Identities Formula

Algebraic identities are equalities true for all permitted values of their variables.

The Formula

(a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2

When to use: Identities are always-true shortcuts — no matter what values you substitute, both sides will always be equal.

Quick Example

(a+b)2=a2+2ab+b2(a+b)^2 = a^2+2ab+b^2 — expand (x+3)2=x2+6x+9(x+3)^2 = x^2 + 6x + 9 directly using this identity.

Notation

equivequiv is sometimes used to denote identity.

What This Formula Means

Algebraic identities are equalities true for all permitted values of their variables.

Identities are always-true shortcuts — no matter what values you substitute, both sides will always be equal.

Formal View

An identity is a statement f(x)equivg(x)f(x)equiv g(x) for all xx in a domain DD.

Worked Examples

Example 1

easy
Expand (x+3)2(x + 3)^2 using the identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2.

Answer

x2+6x+9x^2 + 6x + 9

First step

1
Step 1: a=xa = x, b=3b = 3.

Full solution

  1. 2
    Step 2: (x+3)2=x2+2(x)(3)+32=x2+6x+9(x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9.
  2. 3
    Check: (x+3)(x+3)=x2+3x+3x+9=x2+6x+9(x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9 ✓
Algebraic identities are equations that hold for all values of the variables. The perfect square identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 is faster than FOIL for squaring binomials.

Example 2

medium
Compute 47247^2 mentally using (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 with a=50,b=3a = 50, b = 3.

Example 3

medium
Factor x4−16x^4 - 16 completely.

Common Mistakes

  • Dropping the middle term: writing (a+b)2=a2+b2(a+b)^2=a^2+b^2 - the perfect-square identity has 2ab2ab: (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2
  • Trying to 'solve' an identity for a unique value - every value satisfies it, so there is nothing to solve
  • Mismatching the sign pattern - a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b), but a2+b2a^2+b^2 does not factor over the reals

Why This Formula Matters

Identities are the reusable shortcuts that make expansion and factoring fast and exact; recognizing a2−b2a^2-b^2 or a perfect-square trinomial on sight is what separates fluent algebra from grinding every product by hand. Recognizing it by "Is this equality true for every value of the variable (an always-true pattern), rather than only for special values?" — rather than by familiar numbers — is what lets a student tell it apart from equation (conditional) and evaluating and equivalence transformation in a mixed problem set.

Frequently Asked Questions

What is the Algebraic Identities formula?

Algebraic identities are equalities true for all permitted values of their variables.

How do you use the Algebraic Identities formula?

Identities are always-true shortcuts — no matter what values you substitute, both sides will always be equal.

What do the symbols mean in the Algebraic Identities formula?

equivequiv is sometimes used to denote identity.

Why is the Algebraic Identities formula important in Math?

Identities are the reusable shortcuts that make expansion and factoring fast and exact; recognizing a2−b2a^2-b^2 or a perfect-square trinomial on sight is what separates fluent algebra from grinding every product by hand. Recognizing it by "Is this equality true for every value of the variable (an always-true pattern), rather than only for special values?" — rather than by familiar numbers — is what lets a student tell it apart from equation (conditional) and evaluating and equivalence transformation in a mixed problem set.

What do students get wrong about Algebraic Identities?

The procedure for algebraic identities is the easy part; the trap is dropping the middle term: writing (a+b)2=a2+b2(a+b)^2=a^2+b^2. Asking "Is this equality true for every value of the variable (an always-true pattern), rather than only for special values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Algebraic Identities formula?

Before studying the Algebraic Identities formula, you should understand: variable as generalization, identity vs equation, algebraic pattern.

← Back to Algebraic Identities See All Examples Practice Problems