Difference of Squares

Why a difference of squares always factors

The claim is that $a^2 - b^2$ factors as $(a + b)(a - b)$. There is nothing to guess and nothing to memorize on faith, because you can check it directly. To test whether two factors really rebuild an expression, multiply them out and see what survives.

The same fact has a picture. Start with a square of side $a$, whose area is $a^2$, and cut a smaller square of side $b$ out of one corner, removing area $b^2$. What is left is an L-shaped region of area $a^2 - b^2$. Slice that L into two rectangles and slide them together, and they line up into a single rectangle with sides $a + b$ and $a - b$. No area is created or lost in the move, so the leftover $a^2 - b^2$ must equal the rectangle’s area $(a + b)(a - b)$.

Recognizing a difference of squares

Before you can use the pattern you have to spot it, and the test has three parts. The expression must have exactly two terms. Both terms must be perfect squares, meaning each one is something squared. Each one may be a number like $9 = 3^2$ or $25 = 5^2$, a variable power like $x^2 = (x)^2$ or $x^4 = (x^2)^2$, or a whole term like $9x^2 = (3x)^2$. And the two squares must be joined by a minus sign, one square subtracted from the other. When all three conditions hold, the expression is a difference of squares and factors as the sum times the difference of the two roots.

The minus sign is not a formality. A sum of two squares, like $x^2 + 9$, passes the first two conditions but fails the third. A sum like $x^2 + 9$ does not factor at all over the real numbers. That is not a hole in your method; there is genuinely no pair of real factors whose product is $x^2 + 9$. A later chapter, once numbers beyond the real line are on the table, comes back to this case. For now, read a plus sign between two squares as a full stop and leave the expression alone.

Factoring when the terms carry coefficients

Nothing in the pattern requires the squares to be bare variables. A term like $9x^2$ is a perfect square because $9x^2 = (3x)^2$, so a difference such as $9x^2 - 16$ factors exactly the way $x^2 - 16$ does. The one new skill is taking the square root of each term correctly, coefficient and all. The root of $9x^2$ is $3x$, not $9x$, because you have to square both the $3$ and the $x$ to rebuild $9x^2$.

The routine is always the same. Take the square root of the first term to get $A$, take the square root of the second to get $B$, and write $(A + B)(A - B)$. Then expand it in your head as a check: the two outer squares should return and the middle terms should cancel.

Factoring more than once

Factoring is not finished until nothing that remains can be factored again. Sometimes one pass of the difference-of-squares rule leaves a factor that is itself a difference of squares, and then you apply the rule a second time. This comes up often with fourth powers, because $x^4 = (x^2)^2$ is a perfect square whose root $x^2$ can be a difference of squares in its own right.

Pulling out a common factor first

An expression can hide a difference of squares behind a common factor. Take $2x^2 - 8$. Neither $2x^2$ nor $8$ is a perfect square, so at first glance the pattern does not apply. But the two terms share a factor of $2$, and once you pull it out, what is left is a clean difference of squares:

The habit to build is to factor out the greatest common factor first, then check whether what remains fits a special pattern. Skip that step and you either miss the difference of squares entirely or leave the factoring half-done.

Solving equations and a mental-math shortcut

The pattern turns certain equations into one-liners. To solve $x^2 - 9 = 0$, factor the left side and use the zero-product property, which says a product is zero only when one of its factors is zero:

The same reasoning handles $a^2 - b^2 = 0$ in general. Factoring gives $(a + b)(a - b) = 0$, so either $a + b = 0$ or $a - b = 0$, which means $a = -b$ or $a = b$. A difference of squares is zero exactly when the two quantities are equal or are exact opposites.

The identity is also a real arithmetic shortcut. A product of two numbers that sit the same distance on either side of a round number is a difference of squares in disguise. To multiply $51 \times 49$, notice that $51 = 50 + 1$ and $49 = 50 - 1$, so the product is $(50 + 1)(50 - 1)$:

You have replaced a two-digit multiplication with a square and a subtraction. The same move gives $43 \times 37 = 40^2 - 3^2 = 1600 - 9 = 1591$. This pairing of a sum with a matching difference is worth holding onto for one more reason. The very same $(a + b)(a - b)$ trick is what later lets you clear square roots out of a denominator when you simplify radical expressions.

Common mistakes