Algebraic Identities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Algebraic Identities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An identity holds universally for all valid values, unlike an equation that holds only for specific values.

Common stuck point: Students sometimes treat identities as equations to solve — but they hold for ALL values, so there is nothing to solve.

Sense of Study hint: Test two different values to build intuition, then justify symbolically.

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: a = x, b = 3.
  2. 2
    Step 2: (x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9.
  3. 3
    Check: (x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9 ✓

Answer

x^2 + 6x + 9
Algebraic identities are equations that hold for all values of the variables. The perfect square identity (a+b)^2 = a^2 + 2ab + b^2 is faster than FOIL for squaring binomials.

Example 2

medium
Compute 47^2 mentally using (a - b)^2 = a^2 - 2ab + b^2 with a = 50, b = 3.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Expand (2x - 5)^2.

Example 2

hard
Factor x^3 + 8 using the sum of cubes identity.
← Back to Algebraic Identities Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

variable as generalizationidentity vs equationalgebraic pattern