Difference of Squares

Why the middle terms cancel

Multiply $(a + b)(a - b)$ with FOIL and watch what happens: $a \cdot a = a^2$, then $a \cdot (-b) = -ab$, then $b \cdot a = +ab$, then $b \cdot (-b) = -b^2$. The two middle terms, $-ab$ and $+ab$, are opposites — they cancel completely. What's left is just $a^2 - b^2$.

That cancellation is the whole trick. It only happens because the two binomials have the same two terms with opposite signs between them. $(x + 4)(x - 4)$ cancels; $(x + 4)(x - 3)$ does not.

How to recognize one

An expression is a difference of squares when it passes three checks: exactly two terms, a subtraction between them, and both terms are perfect squares. $x^2 - 25$ passes: $x^2$ is a square and $25 = 5^2$. So does $9x^2 - 16$, because $9x^2 = (3x)^2$ and $16 = 4^2$.

To factor, take the square root of each term and write the answer twice — once with a plus, once with a minus: $x^2 - 25 = (x + 5)(x - 5)$. The order of the factors doesn't matter.

Watch for a common factor first. $3x^2 - 48$ doesn't look like two perfect squares, but pulling out the $3$ gives $3(x^2 - 16)$, and now the inside factors: $3(x + 4)(x - 4)$. Always check for a greatest common factor before checking the pattern.

A sum of squares does not factor

The pattern needs the minus sign. $x^2 + 25$ is a sum of squares, and it does not factor over the real numbers — there is no pair of binomials with real coefficients that multiplies to it. If you try $(x + 5)(x + 5)$ you get $x^2 + 10x + 25$, which has a middle term. On a multiple-choice question, $x^2 + b^2$ staying unfactored is often the point being tested.

Worked examples

Try one yourself

Common questions