Difference of Squares

Spot the pattern A^2 - B^2 and split it in two.

The idea

The difference of squares is a factoring identity: one square minus another always splits into two factors.

Theorem (Difference of squares).

For any two quantities $A$ and $B$, $A^{2} - B^{2} = (A - B)(A + B).$

The identity comes from multiplying a sum by the matching difference: expanding gives $(A + B)(A - B) = A^{2} - AB + AB - B^{2},$ and the two cross terms cancel, leaving only the two squares.

To apply the identity, recognise an expression as one square minus another, name the two quantities $A$ and $B$ being squared, and write down the two factors. Neither $A$ nor $B$ need be a single letter or a plain number: any expression whose square matches the term serves, so a leading coefficient that is itself a perfect square belongs inside $A$ rather than in front.

The cancellation depends on the minus sign, which is what gives the cross terms opposite signs. A sum of squares, $A^{2} + B^{2}$, has no such factorization over the real numbers.

Ways to work on it

• Walkthrough. We factor a difference of two squares step by step.
• Proof. See why it factors — cut a square and rearrange.
• Practice. Two steps. Random x^2 - n^2 with n in 3, 4, , 12.
• Hardest. One step. Leading coefficient ≠ 1.

Not sure where to start? Take the ten-question placement test.