Algebraic Identities — List, Proofs, and Examples

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. $(a+b)^2 = a^2 + 2ab + b^2$ is true everywhere. That "everywhere" is the defining feature.

A quick term, since the rest of the article uses it: a binomial is a two-term expression like $a + b$ or $x - 7$. Most of the core identities are about what happens when you square, cube, or multiply binomials.

This is the narrower, all-values cousin of the general toolkit in our algebraic formula article — every identity is a formula, but a solving formula like the quadratic formula is not an identity, because it only applies to one shape of equation.

The standard algebraic identities list

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

Two-variable square identities

Cube identities

Three-variable identity

The cube expansions are explored further in cube of a binomial, and the general pattern behind all of these — the coefficients $1, 2, 1$ and $1, 3, 3, 1$ — comes from the binomial theorem and Pascal's triangle.

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.

Geometric proof of $(a+b)^2 = a^2 + 2ab + b^2$. Draw a square with side $a + b$. Its total area is $(a+b)^2$. Now split it (see the figure above) into four pieces: a square of area $a^2$, a square of area $b^2$, and two rectangles each of area $ab$. The pieces must sum to the whole:

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

Algebraic proof of $a^2 - b^2 = (a+b)(a-b)$. Expand the right side:

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

The $-ab$ and $+ab$ cancel:

$= a^2 - b^2$

Right side equals left side, so the identity holds.

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 $(a+b)^2 = a^2 + 2ab + b^2$ with $a = x$, $b = 7$.

$(x+7)^2 = x^2 + 2(x)(7) + 7^2$

$= x^2 + 14x + 49$

Final answer: $x^2 + 14x + 49$.

Example 2

Expand $(3y - 5)^2$.

A frequent first move is to write $(3y - 5)^2 = 9y^2 - 25$, squaring each term and subtracting.

Why that is wrong. That skips the middle term entirely and treats the expression like a difference of squares, which it is not. $(3y-5)^2$ is a square of a difference, so the $-2ab$ term must appear.

The correct method. Use $(a-b)^2 = a^2 - 2ab + b^2$ with $a = 3y$, $b = 5$:

$(3y-5)^2 = (3y)^2 - 2(3y)(5) + 5^2$

$= 9y^2 - 30y + 25$

Final answer: $9y^2 - 30y + 25$. The $-30y$ is exactly the term the wrong path drops.

Example 3

Factor $x^2 - 81$.

Recognise a difference of squares: $81 = 9^2$.

$x^2 - 81 = x^2 - 9^2 = (x+9)(x-9)$

Final answer: $(x+9)(x-9)$.

Example 4

Use $(x+a)(x+b) = x^2 + (a+b)x + ab$ to expand $(x+4)(x+6)$.

Here $a = 4$, $b = 6$.

$(x+4)(x+6) = x^2 + (4+6)x + (4)(6)$

$= x^2 + 10x + 24$

Final answer: $x^2 + 10x + 24$.

Example 5

Evaluate $103^2$ using an identity (no long multiplication).

Write $103 = 100 + 3$ and use $(a+b)^2 = a^2 + 2ab + b^2$:

$103^2 = 100^2 + 2(100)(3) + 3^2$

$= 10000 + 600 + 9 = 10609$

Final answer: $10609$. This is the kind of mental-arithmetic shortcut identities were built for.

Example 6

Factor $27x^3 + 8$ using the sum-of-cubes identity.

Rewrite as cubes: $27x^3 = (3x)^3$, $8 = 2^3$. So $a = 3x$, $b = 2$.

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

$27x^3 + 8 = (3x + 2)\big((3x)^2 - (3x)(2) + 2^2\big)$

$= (3x + 2)(9x^2 - 6x + 4)$

Final answer: $(3x+2)(9x^2 - 6x + 4)$.

Where Algebraic Identities Matter

Identities are not exam decoration — they are the engine of fast, reliable manipulation. The reason a student learns them once and reuses them forever is that they convert a hard operation (multiplying out, factoring) into pattern recognition.

• Mental arithmetic: $103^2$, $98 \times 102$, and similar become one-line computations once you see the identity hiding inside.

• Factoring and simplifying: Nearly every factoring technique in later algebra — and every cancellation in a rational expression — starts by spotting a difference of squares or a perfect-square trinomial.

• Calculus and beyond: Differences of cubes and squares appear constantly when simplifying limits and derivatives; the identity is what makes the cancellation legal.

The systematic treatment of expressions as objects you could manipulate by rule traces back to the Persian scholar al-Khwarizmi in the 9th century, whose name gave us algebra; see MacTutor for the history. An identity is that idea distilled: a manipulation guaranteed to be valid every single time.

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: $(a+b)^2 = a^2 + b^2$.

The correct way: $(a+b)^2 = a^2 + 2ab + b^2$. The first instinct on a squared binomial is to square each term and stop — the cross term $2ab$ is the one that gets dropped, more reliably than any other error in algebra. If you remember the four-tile square picture, the two $ab$ rectangles will keep reminding you.

Mistake 2: Confusing $(a-b)^2$ with $a^2 - b^2$

Where it slips in: A squared difference and a difference of squares look similar and get swapped. The memorizer who learned the names without the structure is most at risk.

Don't do this: Treat $(x-3)^2$ as $x^2 - 9$.

The correct way: $(x-3)^2 = x^2 - 6x + 9$ (a squared difference, three terms). $x^2 - 9 = (x+3)(x-3)$ (a difference of squares, factored). They are different identities entirely — one expands, one factors. Read the expression's shape before reaching for a name.

Mistake 3: Sign errors in the cube formulas

Where it slips in: $a^3 - b^3$ versus $a^3 + b^3$. The signs inside the quadratic factor catch people out.

Don't do this: Write $a^3 - b^3 = (a-b)(a^2 - ab + b^2)$.

The correct way: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ — the middle sign in the trinomial is plus. A memory anchor that fixes this: the trinomial's middle term sign is the opposite of the binomial's sign. .

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 $2ab$ middle term and confusing a squared difference $(a-b)^2$ with a difference of squares $a^2 - b^2$.

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

Practice These to Solidify Your Understanding

Work through each, then check the linked sections if you stall.

1. Expand $(2m + 5)^2$. (Revisit Example 1 and Mistake 1.)

2. Factor $49x^2 - 16$. (Revisit Example 3.)

3. Factor $x^3 - 125$ using the difference-of-cubes identity. (Revisit Example 6 and Mistake 3.)

To go deeper with a teacher, explore Bhanzu's algebra tutor or browse the algebra classes that drill this pattern-matching until it is automatic. Want a live Bhanzu trainer to walk through more identity problems? Book a free demo class.

Read More

• Perfect square trinomial — the $(a\pm b)^2$ identity recognised in reverse.

• Polynomials — the expressions identities expand and factor.

• Multiplying binomials — the FOIL view that the square and product identities shortcut.

• Standard form of a quadratic equation — where perfect-square identities help complete the square.

• Sigma notation — compact notation for the kind of sums identities produce.