Difference of Two Squares
A difference of two squares, a2-b2, always factors as (a+b)(a-b). Spotting this pattern turns messy-looking expressions and multiplications into quick, exact work.
The pattern exists because of what subtraction of squares actually measures. Picture a big square with side a and a smaller square with side b cut out of its corner. The leftover L-shaped region has area a2-b2. Cut that L-shape into two rectangles and slide them together: you get one rectangle with sides (a+b) and (a-b). Same area, new shape, so a2-b2 = (a+b)(a-b). This only works for a minus: a2+b2 has no such rearrangement, because you can't carve a square's area away and add it back without canceling, so the sum of two squares stays stuck and does not factor over the real numbers.
To use the pattern you need both terms to be perfect squares, meaning each one is literally something squared. 9x2-12 fails this test: 9x2 is (3x)2, but 12 is not the square of any whole number, so the expression doesn't factor this way at all. Compare that to 49x2-64, where 49x2 = (7x)2 and 64 = 82, so it factors as (7x+8)(7x-8). Setting that equal to zero gives 7x+8=0 or 7x-8=0, so x = -8/7 or x = 8/7.
This pattern is one of the few factoring shortcuts worth memorizing instantly, alongside perfect-square trinomials. It shows up constantly inside harder quadratics and radical equations, where recognizing the hidden squares turns a multi-step solve into one factoring move.
Key terms
- Difference of two squares
- An expression of the form a2-b2, which always factors as (a+b)(a-b).
- Perfect square
- A number or expression that equals something else squared, like 64 = 82 or 9x2 = (3x)2.
- Binomial
- An expression with exactly two terms, such as a+b or 7x-8.
- Factoring over the integers
- Breaking an expression into factors whose coefficients and constants are all integers.
Watch it explained
- How it worksHow does slicing a small square out of a big one and rearranging the leftover strip prove a2-b2 equals (a+b)(a-b)?
- Common mistakeWhy isn't 9x2-12 a difference of two squares, given that 9x2 is a perfect square?
- Worked exampleFind the factorization of 49x2-64 and use it to get both solutions of 49x2-64=0.
- Worked exampleFind 103 times 97 in your head by reading it as a difference of two squares.
Check yourself
End of topic.