How this instrument works
Whenever one square is subtracted from another, a² − b² can be rewritten instantly as (a − b)(a + b). Multiply that pair back out and the two cross terms cancel on their own: one comes out positive, the other negative, both built from the identical two starting values, so they erase each other and leave only the two squared amounts with a subtraction sitting between them. That cancellation is the entire mechanism behind the identity, and it holds for every pair of real numbers, not merely tidy ones.
This sheet stays narrowly focused on that one pattern rather than trying to be a general factoring tool. A page elsewhere on this site finds the greatest common factor of two whole numbers using the Euclidean algorithm, repeatedly dividing and taking remainders until nothing is left to share — a method built for pairs with no special algebraic shape at all. The pattern here needs no such search: a subtraction of two squares announces its own factor pair on sight, the moment it is recognized, with none of the trial division that ordinary factoring often demands.
Two edge cases show the identity's reach. Let the second number fall to zero and the whole expression collapses to a plain square, a² − 0, which factors as a times a — the two factors becoming identical rather than opposite. Let the two starting numbers match instead, so a equals b, and a² − b² collapses the other direction, straight to zero: one of the two factors is forced to 0, and multiplying it against any second factor, however large, still returns nothing.
- Enter the first number into a, from a² − b².
- Enter the second number into b — the value being squared and subtracted.
- Read First factor for a − b, the smaller of the pair.
- Read Second factor for a + b, the larger of the pair whenever both a and b are positive.
- Multiply First factor by Second factor by hand if you want to confirm the product lands back on a² − b².
Worked example — factoring 5² − 3²
Set a=5 and b=3. Directly, 5² − 3² = 25 − 9 = 16. The identity gets there without squaring anything: First factor is 5 − 3 = 2, and Second factor is 5 + 3 = 8, and 2 × 8 = 16 — the same answer, reached without ever computing 25 or 9 along the way.
Two further pairs show the pattern's edges. With a=10 and b=0, 10² − 0² = 100, and the factors come out as 10 and 10 — a plain square rather than a genuine difference, since subtracting zero changes nothing. With a=7 and b=7, 7² − 7² = 0, and First factor drops to 0 while Second factor still reads 14; multiplying 0 by 14 gives 0 regardless of how large that second number is, since a zero factor forces a zero product no matter what stands beside it.
Questions
What is the difference-of-squares identity?
It is the statement that a² − b² always equals (a − b)(a + b), for any real numbers a and b. Subtracting 3² from 5² gives 25 − 9 = 16, and the identity reaches the same 16 through 2 × 8, since 5 − 3 = 2 and 5 + 3 = 8.
Why does a² − b² factor into (a − b)(a + b)?
Expand (a − b)(a + b) term by term and the cross terms, ab and −ab, cancel each other exactly, leaving only a² − b². That cancellation happens for every pair of numbers, which is why the rewrite is a genuine identity rather than something that only works for particular values.
What happens when b is zero?
The expression reduces to a plain square, a² − 0, and the two factors become a − 0 and a + 0 — both equal to a itself. With a=10 and b=0, both First factor and Second factor read 10, and their product, 100, matches 10² exactly.
What happens when a and b are equal?
First factor drops to exactly 0, since a − a is always 0, while Second factor still reads a + a. With a=7 and b=7, First factor is 0 and Second factor is 14; the product is 0 regardless of how large Second factor happens to be, because multiplying anything by 0 gives 0.
How is this different from general polynomial factoring?
General factoring covers many shapes and often needs trial-and-error to find a matching pair of factors. This sheet handles exactly one recognizable shape — a subtraction of two squares — and returns its factors immediately from a and b, with none of the guessing that broader factoring methods require.
Does this identity work on plain numbers as well as algebraic expressions?
Yes — a and b can stand for any real numbers, not only variables in an equation. Treating 5 and 3 as ordinary numbers rather than symbols still gives 5² − 3² = (5−3)(5+3) = 2 × 8 = 16, exactly as the algebraic version predicts.