x² + 3x − 28 factors into (x + 7)(x − 4) — two numbers with opposite signs that multiply to −28. These free printable worksheets drill the case teachers see most.
When the last term is negative, the two numbers have opposite signs: one plus, one minus. That is the case a test reaches for most, and the one where a child is likeliest to attach the wrong sign to the bigger number. They multiply to a negative and add to the middle term, and which one is bigger is decided by the sign of b.
Why this sheet works
So for x² + 3x − 28 the pair multiplies to −28 and adds to +3, which is +7 and −4 — the larger number takes the sign of b. Swap them to −7 and +4 and the product is still −28 but the sum is −3, and the whole answer is wrong by a sign. The walks decide the size first and the signs second, so that swap cannot slip through.
How this one works
One question from this sheet, worked through a step at a time.
x² − 15x − 100
x² − 15x − 100
multiply to−100· add to−15
−100 < 0→one +, one −
1·1002·504·255·2010·10
1·1002·504·255·2010·10
−20and5
x² − 15x − 100=(x − 20)(x + 5)
Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to −100, the last term, and they ADD to −15, the number in front of x.
−100 is negative, and a product is only negative when the two numbers have OPPOSITE signs. So one is positive and one is negative, and the bigger of the two takes the sign of −15.
Now list the pairs that multiply to 100. There are only 5, so this is a short search rather than a guess: 1 and 100, 2 and 50, 4 and 25, 5 and 20, 10 and 10.
Which of them adds to −15? −20 × 5 = −100 and −20 + 5 = −15. So the two numbers are −20 and 5.
Put them in the brackets: x² − 15x − 100 = (x − 20)(x + 5). Multiply it back out if you want to check — the x terms should add up to −15x again.
