x² − 9x + 20 factors into (x − 4)(x − 5) — two negative numbers that multiply to 20 and add to −9. These free printable worksheets drill the both-negative case on its own.
Factoring x² + bx + c is a search for two numbers that multiply to c and add to b, and this sheet fixes both signs so the search has a shape. When c is positive and b is negative, the two numbers are both negative: they multiply to a positive and add to a negative. Knowing that before you start turns a hunt into a short check of a few pairs.
Why this sheet works
The middle term is where the sign lands. Both numbers of the pair are negative, so (x − 4)(x − 5) gives −4x and −5x, which add to the −9x in the middle, and the +20 is negative times negative. A child who writes (x + 4)(x + 5) has the product right and the sum wrong, which is exactly the mistake this narrow sheet is built to surface.
How this one works
One question from this sheet, worked through a step at a time.
x² − 32x + 231
x² − 32x + 231
multiply to231· add to−32
231 > 0→same sign
−32 < 0→both −
1·2313·777·3311·21
1·2313·777·3311·21
−21and−11
x² − 32x + 231=(x − 21)(x − 11)
Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to 231, the last term, and they ADD to −32, the number in front of x.
231 is positive, so the two numbers share a sign — a product is only positive that way. And −32 is negative, so the sign they share is negative: two negatives multiply to a positive and add to a negative.
Now list the pairs that multiply to 231. There are only 4, so this is a short search rather than a guess: 1 and 231, 3 and 77, 7 and 33, 11 and 21.
Which of them adds to −32? −21 × −11 = 231 and −21 + −11 = −32. So the two numbers are −21 and −11.
Put them in the brackets: x² − 32x + 231 = (x − 21)(x − 11). Multiply it back out if you want to check — the x terms should add up to −32x again.
