Factoring x² − bx + c, step by step
x² − 9x + 20 factors into (x − 4)(x − 5) — two negative numbers that multiply to 20 and add to −9.
x² − 29x + 138
- Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to 138, the last term, and they ADD to −29, the number in front of x.
- 138 is positive, so the two numbers share a sign — a product is only positive that way. And −29 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 138. There are only 4, so this is a short search rather than a guess: 1 and 138, 2 and 69, 3 and 46, 6 and 23.
- Which of them adds to −29? −6 × −23 = 138 and −6 + −23 = −29. So the two numbers are −6 and −23.
- Put them in the brackets: x² − 29x + 138 = (x − 6)(x − 23). Multiply it back out if you want to check — the x terms should add up to −29x again.
How it works.
- 01
Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to 138, the last term, and they ADD to −29, the number in front of x.
- 02
138 is positive, so the two numbers share a sign — a product is only positive that way. And −29 is negative, so the sign they share is negative: two negatives multiply to a positive and add to a negative.
- 03
Now list the pairs that multiply to 138. There are only 4, so this is a short search rather than a guess: 1 and 138, 2 and 69, 3 and 46, 6 and 23.
- 04
Which of them adds to −29? −6 × −23 = 138 and −6 + −23 = −29. So the two numbers are −6 and −23.
- 05
Put them in the brackets: x² − 29x + 138 = (x − 6)(x − 23). Multiply it back out if you want to check — the x terms should add up to −29x again.