Find two numbers that multiply to the last term and add to the middle one. These free printable worksheets drill that one search until it is automatic.
The method is a search, and a search is only teachable once you know what you are searching for. Two conditions decide the pair and both have to hold: they multiply to c and they add to b. A student who checks only the product finds 1 and 12 for x² + 7x + 12 and stops; one who checks only the sum never gets started. Every walk here states both before trying a single number.
Why this sheet works
The signs are three separate lessons, so they are three settings rather than one slider. With c positive the two numbers share a sign and b says which; with c negative they have opposite signs and b says which one is bigger. That is different reasoning, taught a week apart, and mixing it in from the first sheet is what makes factoring feel like guessing.
How this one works
One question from this sheet, worked through a step at a time.
x² + 16x + 28
x² + 16x + 28
multiply to28· add to16
1·282·144·7
1·282·144·7
14and2
x² + 16x + 28=(x + 14)(x + 2)
Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to 28, the last term, and they ADD to 16, the number in front of x.
Now list the pairs that multiply to 28. There are only 3, so this is a short search rather than a guess: 1 and 28, 2 and 14, 4 and 7.
Which of them adds to 16? 14 × 2 = 28 and 14 + 2 = 16. So the two numbers are 14 and 2.
Put them in the brackets: x² + 16x + 28 = (x + 14)(x + 2). Multiply it back out if you want to check — the x terms should add up to 16x again.
