Something squared minus something squared is the sum times the difference — the one factorization with no middle term to hunt for. Free printable worksheets.
Ready for harder? The gap between two squares — the same identity, used to find the two numbers rather than to factorize.
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Ready to printDifference of two squares
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What's on this sheet
The pattern is worth recognizing precisely because it skips the search: no two numbers to find, no middle term to match. But it only works when the expression really is one square minus another, and a student who has learned the shortcut applies it to x² − 20 and to x² + 64 just as happily. So every walk here checks the shape first — is each part a perfect square, and is it a minus — before it factors anything.
Why this sheet works
Multiplying it back out once is the whole proof and it takes two lines: the outer and inner products are +kx and −kx, and they cancel. That is also the answer to “why is there no middle term”, and a student who has watched it disappear does not have to take the rule on trust.
How this one works
One question from this sheet, worked through a step at a time.
x² − 256
x² − 256
x² = (x)²256 = 16²
(x + 16)(x − 16)
+16x−16xcancel
x² − 256
x² − 256=(x + 16)(x − 16)
Before factoring anything, check the shape. There are two parts with a − between them, and each part has to be a perfect square: x² is x squared, and 256 is 16 squared. Both are, and it is a minus — so this is a difference of two squares.
That shape always factors the same way, and there is nothing to search for: it is the SUM times the DIFFERENCE of the two things being squared — (x + 16)(x − 16).
Multiply it back out if you want to see why. The outer pair gives −16x and the inner pair gives +16x; they cancel, which is why there is no middle term left. What survives is x² − 256.
So the missing number is 16: 256 is 16 squared, so the two brackets are (x + 16) and (x − 16).
Each one is a difference of two squares. Fill in the missing number.
1x² − 256=(x + )(x − )
29x² − 400=(x + 20)(x − 20)
381x² − 289=(x + 17)(x − 17)
4121x² − 784=(x + 28)(x − 28)
5x² − 64=(x + )(x − )
6121x² − 100=(x + 10)(x − 10)
7121x² − 961=(x + 31)(x − 31)
8x² − 576=(x + )(x − )
9x² − 100=(x + )(x − )
1049x² − 484=(x + 22)(x − 22)
Difference of two squares · Both formsAnswer key
1. x² − 256 = (x + 16)(x − 16) (256 is 16 squared, so the two brackets are (x + 16) and (x − 16))2. 9x² − 400 = (3x + 20)(3x − 20) (9x² is 3x squared, so the two brackets start 3x rather than 9x)3. 81x² − 289 = (9x + 17)(9x − 17) (81x² is 9x squared, so the two brackets start 9x rather than 81x)4. 121x² − 784 = (11x + 28)(11x − 28) (121x² is 11x squared, so the two brackets start 11x rather than 121x)5. x² − 64 = (x + 8)(x − 8) (64 is 8 squared, so the two brackets are (x + 8) and (x − 8))6. 121x² − 100 = (11x + 10)(11x − 10) (121x² is 11x squared, so the two brackets start 11x rather than 121x)7. 121x² − 961 = (11x + 31)(11x − 31) (121x² is 11x squared, so the two brackets start 11x rather than 121x)8. x² − 576 = (x + 24)(x − 24) (576 is 24 squared, so the two brackets are (x + 24) and (x − 24))9. x² − 100 = (x + 10)(x − 10) (100 is 10 squared, so the two brackets are (x + 10) and (x − 10))10. 49x² − 484 = (7x + 22)(7x − 22) (49x² is 7x squared, so the two brackets start 7x rather than 49x)
