Difference of Squares Formula a2 - b2 with Examples

Key Points

• The difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$, factoring a subtraction of two squares into two brackets.

• Expanding $(a+b)(a-b)$ cancels the two middle terms, $-ab$ and $+ab$, which is why no middle term survives.

• The sum of squares $a^2 + b^2$ does not factor over the real numbers, so the pattern works only for subtraction.

• The formula turns awkward multiplications into easy ones, since $97 \times 103$ becomes $100^2 - 3^2$.

• Either $a$ or $b$ can itself be a compound expression, so $x^4 - 16$ factors twice by applying the pattern again.

What Is the Difference of Squares Formula?

The difference of squares formula factors any expression that is one square subtracted from another:

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

It is one member of the standard family of algebraic identities. This article treats that one identity in depth; the algebraic identities page sets out the family together.

Why the Middle Terms Vanish

Multiply the two brackets out term by term and the structure becomes obvious:

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

The two middle terms are $-ab$ and $+ba$. Since $ab = ba$, they are equal in size and opposite in sign, so they cancel exactly and leave $a^2 - b^2$.

That cancellation is the whole mechanism, and it is also why the sum of squares does not factor this way. To get $a^2 + b^2$ the middle terms would still have to cancel while the last term came out positive, which would need $-b^2$ to become $+b^2$. No pair of real brackets does that.

Variable Key

Examples of the Difference of Squares Formula

Example 1

Factor $x^2 - 49$.

Here $a = x$ and $b = 7$, since $49 = 7^2$:

$$x^2 - 49 = (x+7)(x-7)$$

Final answer: $(x+7)(x-7)$

Example 2

Factor $x^2 + 25$.

The pattern is fresh in mind, so applying it gives $(x+5)(x-5)$. Expand that to check:

$$(x+5)(x-5) = x^2 - 25$$

That is $x^2 - 25$, not $x^2 + 25$. The sign of the final term came out negative, because the identity only ever produces a subtraction.

A sum of two squares has no factorisation over the real numbers, so $x^2 + 25$ is already in its simplest real form.

Final answer: $x^2 + 25$ does not factor over the reals. The difference of squares pattern requires a minus sign between the terms.

Example 3

Factor $9y^2 - 64$.

Both terms are squares, with $9y^2 = (3y)^2$ and $64 = 8^2$:

$$9y^2 - 64 = (3y+8)(3y-8)$$

Final answer: $(3y+8)(3y-8)$

Example 4

Factor $x^4 - 16$ completely.

First pass takes $a = x^2$ and $b = 4$:

$$x^4 - 16 = (x^2+4)(x^2-4)$$

The second bracket is itself a difference of squares, so apply the pattern again:

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

Final answer: $(x^2+4)(x+2)(x-2)$

The first bracket is a sum of squares and stops there, which is the signal that the factorisation is complete over the reals.

Example 5

Compute $97 \times 103$ mentally.

Both numbers sit 3 away from 100, so write them as $100 - 3$ and $100 + 3$:

$$97 \times 103 = (100-3)(100+3) = 100^2 - 3^2 = 10000 - 9 = 9991$$

Final answer: 9,991

Any pair of numbers equally spaced either side of a round figure multiplies this way, which turns a two-digit multiplication into a subtraction.

Don't Confuse This With the Other Square Identities

The first row is the only one of the four that factors a subtraction into two different brackets. The two squared-bracket rows produce a middle term rather than losing one, which is the quickest way to tell them apart.

Where Do Students Trip Up on the Difference of Squares?

Mistake 1: Applying the pattern to a sum of squares

Where it slips in: The two terms are squares and the formula is memorable, so the sign between them gets overlooked.

Don't do this: $x^2 + 25 = (x+5)(x-5)$

The correct way: Check the sign first. Only $a^2 - b^2$ factors this way, and $(x+5)(x-5)$ expands back to $x^2 - 25$.

Students who expand their own answer before moving on catch this immediately, because the expansion returns a minus sign that was not in the question. Expanding a factorisation is a complete check, not a partial one, and it costs one line.

Mistake 2: Missing a common factor first

Where it slips in: The expression looks like a difference of squares, so the pattern gets applied before anything else.

Don't do this: Factor $2x^2 - 18$ as $(\sqrt{2}x+\sqrt{18})(\sqrt{2}x-\sqrt{18})$.

The correct way: Take out the common factor 2 first, giving $2(x^2-9) = 2(x+3)(x-3)$. Checking for a common factor before any identity keeps the brackets free of surds.

Mistake 3: Stopping after one application

Where it slips in: The first factorisation looks finished, and the resulting brackets are not re-examined.

Don't do this: Leave $x^4 - 16$ as $(x^2+4)(x^2-4)$.

The correct way: Test every bracket again. The $x^2-4$ factors further into $(x+2)(x-2)$, and only a sum of squares or a non-square tells you to stop.

Key Takeaways

• The difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$.

• The middle terms cancel because $-ab$ and $+ba$ are equal and opposite, which is the whole mechanism.

• A sum of squares does not factor over the reals, so the minus sign is a precondition rather than a detail.

• Check for a common factor first, then apply the pattern repeatedly until only sums of squares remain.

• To work through algebraic identities with a teacher, explore an algebra tutor or math classes online.

A Practical Next Step

Factor $16x^2 - 81$ and $x^4 - 1$ completely. Then compute $48 \times 52$ mentally using the identity, and expand each answer to confirm it.

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Read More

• A Plus B Whole Square

• Factorization of Algebraic Expressions

• Perfect Squares

• Completing the Square

• Perfect Square Trinomial