(x + 8)(x − 8) is a sum times a difference, so the answer is x² − 64 with no middle term to work out. These free printable worksheets read the difference of two squares in the direction that saves time.
Most factoring practice runs one way: here is x² − 64, split it into brackets. This page runs the other way. The brackets are handed over already written — the same two things, a plus in one and a minus in the other — and the job is to see what that shape gives you before multiplying anything.
Why this sheet works
What it gives you is the middle term, for free. Multiply all four pairs of (x + 8)(x − 8) and you get x² + 8x − 8x − 64; the two middle products are the same size with opposite signs, so they cancel every time. A child who recognizes the shape writes x² − 64 in one step. A child who does not spends four multiplications getting to the same place, and has two more chances to slip a sign.
How this one works
One question from this sheet, worked through a step at a time.
(x + 20)(x − 20) + 18
(x + 20)(x − 20)=x² − 400
x² − 400 + 18
−400 + 18 = −382
(x + 20)(x − 20) + 18=x² − 382
The two brackets hold the same two things — x and 20 — with a + in one and a − in the other. That is a sum times a difference, so the product is the difference of their squares: no middle term to work out.
20 squared is 400, so the brackets come to x² − 400. Now the 18 that was waiting outside them: −400 + 18 is −382.
So (x + 20)(x − 20) + 18 is x² − 382: the brackets multiply to x² − 400, and −400 + 18 is −382.
