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  • a²x² − b², whole roots

Difference of squares with coefficients worksheets

9x² − 64 is still one square minus another — but now the square in front matters. These free printable worksheets factor a difference of squares that carries a coefficient on x².

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Ready to printDifference of squares with coefficients
Difference of squares with coefficients worksheet — a²x² − b², whole roots, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The pattern is the same as the plain case, sum times difference, but the root a child takes is the harder one. √(9x²) is 3x — not 9x, and not 3 — because it is the root of the number and the root of x² taken together. That single step is what this sheet exists to drill, and it is the one students skip.

Why this sheet works

So 9x² − 64 factors as (3x + 8)(3x − 8), and checking it means multiplying back out until the middle terms cancel. Seeing +24x and −24x disappear is the whole proof that there is no middle term, and it also catches the child who wrote 9x by mistake, because then the outer and inner products would not cancel.

How this one works

One question from this sheet, worked through a step at a time.

121x² − 1296

121x² − 1296
121x² = (11x)²1296 = 36²
(11x + 36)(11x − 36)
+396x−396xcancel
121x² − 1296
121x² − 1296=(11x + 36)(11x − 36)
  1. Before factoring anything, check the shape. There are two parts with a − between them, and each part has to be a perfect square: 121x² is 11x squared, and 1296 is 36 squared. Both are, and it is a minus — so this is a difference of two squares.
  2. 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 — (11x + 36)(11x − 36).
  3. Multiply it back out if you want to see why. The outer pair gives −396x and the inner pair gives +396x; they cancel, which is why there is no middle term left. What survives is 121x² − 1296.
  4. So the missing number is 11, not 121: 121x² is 11x squared, so the two brackets start 11x rather than 121x. The square root of 121x² is 11x — the root of the number AND the root of x², taken together.

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