Book A Free Math Class

# Algebraic Identities — List, Proofs, and Examples

## TL;DR

Algebraic identities are equations that stay true for every value of their variables — for example (a+b)²=a²+2ab+b² and a²−b²=(a+b)(a−b). This article gives you the full standard list (square, cube, and three-variable identities), a geometric and an algebraic proof, six worked examples, and the slips that cost the most marks.

## Algebraic Identities — Definition, List, Proofs, and Examples

**Algebraic identities** are equations that stay true for _every_ value of their variables — for example (a+b)²=a²+2ab+b² and a²−b²=(a+b)(a−b). This article gives you the full standard list (square, cube, and three-variable identities), a geometric and an algebraic **proof**, six worked examples, and the slips that cost the most marks.

## The equation that is true no matter what you put into it

Most equations are fussy. 2x+3=11 is true only when x=4 — change the value and it breaks. But (a+b)²=a²+2ab+b² never breaks. Pick any numbers you like for a and b, and both sides land on the same answer. That unconditional truth is what makes an identity a tool you can lean your whole weight on.

## What is an Algebraic Identity?

An **algebraic identity** is an equation in which the left-hand side equals the right-hand side for _all_ values of the variables involved. Substitute anything, and equality holds.

Contrast that with an ordinary equation, which is only true for particular values. x+2=5 is true at x=3 and false everywhere else. (a+b)²=a²+2ab+b² is true everywhere. That "everywhere" is the defining feature.

## The standard algebraic identities list

Here is the working set, grouped by degree. These are the ones worth knowing cold.

**Two-variable square identities**  
| Identity | Use |  
| --- | --- |  
| (a+b)²=a²+2ab+b² | Expand a squared sum |  
| (a−b)²=a²−2ab+b² | Expand a squared difference |  
| a²−b²=(a+b)(a−b) | Factor a difference of squares |  
| (x+a)(x+b)=x²+(a+b)x+ab | Expand a product of two linear binomials |

**Cube identities**  
| Identity | Use |  
| --- | --- |  
| (a+b)³=a³+3a²b+3ab²+b³ | Expand a cubed sum |  
| (a−b)³=a³−3a²b+3ab²−b³ | Expand a cubed difference |  
| a³+b³=(a+b)(a²−ab+b²) | Factor a sum of cubes |  
| a³−b³=(a−b)(a²+ab+b²) | Factor a difference of cubes |

**Three-variable identity**  
| Identity | Use |  
| --- | --- |  
| (a+b+c)²=a²+b²+c²+2ab+2bc+2ca | Expand a squared trinomial |  
| a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca) | A useful factoring identity |

## How do you prove an Algebraic Identity?

Two routes prove an identity: a **geometric proof** (areas or volumes) and an **algebraic proof** (expanding and collecting terms). Both confirm the two sides are genuinely the same expression.

## Examples of Algebraic Identities

Six problems, easier to harder, each one matching the expression to the right identity before substituting.

### Example 1

**Expand (x+7)².**  
Use (a+b)²=a²+2ab+b² with a=x, b=7.

Final answer: x²+14x+49.

### Example 2

**Expand (3y−5)².**  
Use (a−b)²=a²−2ab+b².

Final answer: 9y²−30y+25.

### Example 3

**Factor x²−81.** 
Recognise a difference of squares: 81=9².

Final answer: (x+9)(x−9).

### Example 4

**Use (x+a)(x+b)=x²+(a+b)x+ab to expand (x+4)(x+6).**

Final answer: x²+10x+24.

### Example 5

**Evaluate 103² using an identity.** 
Use (a+b)²=a²+2ab+b²: 103²=100²+2(100)(3)+3².

Final answer: 10609.

### Example 6

**Factor 27x³+8 using the sum-of-cubes identity.**

Final answer: (3x+2)(9x²−6x+4).

## Common Mistakes With Algebraic Identities

### Mistake 1: Forgetting the middle term

**Where it slips in:** Any time a binomial is squared.  
**Don't do this:** (a+b)²=a²+b².  
**The correct way:** (a+b)²=a²+2ab+b².

### Mistake 2: Confusing (a−b)² with a²−b²

**Where it slips in:** A squared difference and a difference of squares look similar.  
**Don't do this:** Treat (x−3)² as x²−9.  
**The correct way:** (x−3)²=x²−6x+9.

### Mistake 3: Sign errors in the cube formulas

**Where it slips in:** a³−b³ versus a³+b³.  
**Don't do this:** Write a³−b³=(a−b)(a²−ab+b²).  
**The correct way:** a³−b³=(a−b)(a²+ab+b²).

## Conclusion

Algebraic identities are equations true for every value of their variables — that unconditional truth is what separates them from ordinary equations.

The core list is small: three square identities, four cube identities, and a couple of three-variable ones.

Every identity can be proved geometrically (areas) or algebraically (expand and collect).

The most expensive mistakes are dropping the middle term and confusing a squared difference with a difference of squares.

Master these once and they pay off across factoring, mental arithmetic, and calculus for years.

## Practice These to Solidify Your Understanding

1. Expand (2m+5)².  
2. Factor 49x²−16.  
3. Factor x³−125.