Factoring Difference of Squares

The difference of squares is one of the most recognizable factoring patterns in algebra. Once you learn to spot it, you can factor certain binomials instantly — no AC method, no trial and error, just one clean formula.

The pattern is:

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

This works because when you multiply $(a + b)(a - b)$ using FOIL, the middle terms cancel:

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

The key is recognizing when an expression fits this pattern: it must be a binomial (two terms), both terms must be perfect squares, and they must be connected by subtraction.

Recognizing Perfect Squares

Before you can use this pattern, you need to quickly identify perfect squares.

Perfect square numbers: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \ldots$

Perfect square variables: A variable expression is a perfect square when the exponent is even — $x^2$, $x^4$, $x^6$, $y^8$, etc. The “square root” is the variable raised to half the exponent.

Basic Examples

Example 1: Factor $x^2 - 9$

Identify the squares: $x^2 = (x)^2$ and $9 = (3)^2$.

Apply the pattern:

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

Check: $(x + 3)(x - 3) = x^2 - 3x + 3x - 9 = x^2 - 9$ . Correct.

Example 2: Factor $4x^2 - 25$

Identify the squares: $4x^2 = (2x)^2$ and $25 = (5)^2$.

Apply the pattern:

$4x^2 - 25 = (2x)^2 - (5)^2 = (2x + 5)(2x - 5)$

Check: $(2x + 5)(2x - 5) = 4x^2 - 10x + 10x - 25 = 4x^2 - 25$ . Correct.

Example 3: Factor $49a^2 - 64b^2$

Identify the squares: $49a^2 = (7a)^2$ and $64b^2 = (8b)^2$.

Apply the pattern:

$49a^2 - 64b^2 = (7a + 8b)(7a - 8b)$

Higher-Degree Differences of Squares

The pattern works whenever both terms are perfect squares — even with higher exponents.

Example 4: Factor $x^4 - 16$

Identify the squares: $x^4 = (x^2)^2$ and $16 = (4)^2$.

Apply the pattern:

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

But wait — factor completely! The second factor, $x^2 - 4$, is itself a difference of squares:

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

So the complete factorization is:

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

The first factor, $x^2 + 4$, is a sum of squares — it does not factor over the real numbers (more on this below).

Example 5: Factor $81y^4 - 1$

Identify the squares: $81y^4 = (9y^2)^2$ and $1 = (1)^2$.

$81y^4 - 1 = (9y^2 + 1)(9y^2 - 1)$

Factor the second term again:

$9y^2 - 1 = (3y + 1)(3y - 1)$

Complete factorization:

$81y^4 - 1 = (9y^2 + 1)(3y + 1)(3y - 1)$

Factor Out the GCF First

Always check for a GCF before applying the difference of squares pattern.

Example 6: Factor $50x^2 - 18$

Step 1 — Factor out the GCF: The GCF of 50 and 18 is 2.

$50x^2 - 18 = 2(25x^2 - 9)$

Step 2 — Apply the difference of squares pattern: $25x^2 = (5x)^2$ and $9 = (3)^2$.

$2(25x^2 - 9) = 2(5x + 3)(5x - 3)$

Without factoring out the 2 first, you would not see the difference of squares pattern at all, since 50 and 18 are not perfect squares.

Example 7: Factor $3x^3 - 12x$

Step 1 — Factor out the GCF: The GCF is $3x$.

$3x^3 - 12x = 3x(x^2 - 4)$

Step 2 — Apply the difference of squares:

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

The Sum of Squares Does NOT Factor

This is a critical distinction. While $a^2 - b^2$ factors beautifully, the sum of squares $a^2 + b^2$ does not factor over the real numbers.

$a^2 + b^2 \neq (a + b)(a - b) \qquad \text{(wrong!)}$

$a^2 + b^2 \neq (a + b)^2 \qquad \text{(wrong — that equals } a^2 + 2ab + b^2\text{)}$

There is simply no way to write $x^2 + 9$ as a product of two real-number binomials. When you encounter a sum of squares, it is already fully factored (over the reals). Leave it alone.

Quick test: Is it a difference (subtraction) of two perfect squares? If yes, factor it. If it is a sum (addition) of two squares, stop — it does not factor.

Difference of Squares with Subtraction in Disguise

Sometimes the expression is rearranged so the subtraction is not immediately obvious.

Example 8: Factor $-25 + 4x^2$

Rewrite in standard order:

$4x^2 - 25 = (2x + 5)(2x - 5)$

Example 9: Factor $9 - x^2$

This is already a difference of squares — just written with the constant first:

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

You can also write it as $-(x^2 - 9) = -(x + 3)(x - 3)$, but $(3 + x)(3 - x)$ is perfectly valid and more direct.

Real-World Application: Electrician — Power Difference in Parallel Circuits

An electrician compares the power dissipated by a resistive load at two different operating currents. Using the formula $P = I^2 R$, the difference in power between current $I_1$ and current $I_2$ through the same resistance $R$ is:

$\Delta P = I_1^2 R - I_2^2 R$

Factor out $R$, then apply the difference of squares:

$\Delta P = R(I_1^2 - I_2^2) = R(I_1 + I_2)(I_1 - I_2)$

If $I_1 = 8$ amps and $I_2 = 5$ amps with $R = 10$ ohms:

$\Delta P = 10(8 + 5)(8 - 5) = 10 \times 13 \times 3 = 390 \text{ watts}$

The factored form makes the calculation easier — you avoid squaring each current separately. It also reveals that the power difference depends on both the sum and the difference of the currents.

Common Mistakes to Avoid

1. Trying to factor a sum of squares. Remember: $a^2 + b^2$ does not factor over the reals. Only the difference $a^2 - b^2$ factors.
2. Forgetting to factor completely. After applying the pattern, check whether either factor is itself a difference of squares (like $x^4 - 16$).
3. Not factoring out the GCF first. Expressions like $50x^2 - 18$ do not look like a difference of squares until you factor out the 2.
4. Misidentifying perfect squares. Remember that $x^3$ is not a perfect square (odd exponent), and numbers like 12, 18, or 20 are not perfect squares. The coefficient and every variable factor must be perfect squares.
5. Confusing $(a - b)^2$ with $a^2 - b^2$. The expression $(a - b)^2 = a^2 - 2ab + b^2$ — it has three terms. The difference of squares $a^2 - b^2$ has only two terms and no middle term.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The difference of squares pattern is $a^2 - b^2 = (a + b)(a - b)$
• Both terms must be perfect squares connected by subtraction
• The sum of squares $a^2 + b^2$ does not factor over the real numbers
• Always factor out the GCF first — it can reveal a hidden difference of squares
• Always factor completely — check if either result is itself a difference of squares
• This pattern appears frequently in physics, engineering, and trade calculations whenever you need the difference of two squared quantities

Return to Algebra for more topics in this section.