To answer this question, it may help to look ahead to factoring quadratic trinomials (in the form ax2+bx+c).
- Assume a is 1. Now the equation looks like x2+bx+c.
- if a is not 1, see note at the bottom.
- List all the factor pairs of c. For example, if c=24 the factor pairs are:
- 1, 24
- 2, 12
- 3, 8
- 4, 6
stop when you get to the next highest square root, in this case 5=√25 (5 is not a factor of 24, so it's not included on the list)
3. Look at the SIGN (+/-) of c. If (+), then both factors have the same sign (positive times positive is a positive, negative times a negative is a positive). If (-), then the factors have opposite signs.
⇑This is "kinda" a big deal.
4, If c is (+), add the factors in each pair and compare the sum of each pair to |b| (the absolute value of b, its distance from zero. Just ignore the sign).
If c is (-), find the difference of the factors in each pair and compare this number to |b|.
(absolute value, as above)
5, Fill in the blanks (x )(x ). Ignore the operations in the middle of the parentheses for now. Order doesn't matter. For example, x2-10x+24 will factor into (check steps 3 and 4 above) (x _ 6)(x _ 4)
6, Recall the sign of c. Will the signs of the factors be the same or opposite? In the case above, the factors will have the same sign.
7, Look at the sign of b. If the factors of c have the same sign, this is the sign that fills in BOTH blanks. (above, (x-6)(x-4))
8, If the factors of c have opposite signs, the factor further away from zero wins when you add them to get b.
For example, in x2-23x-24 the sign of c is (-), so b is a difference of factors that multiply to 24. 1 and 24 have a difference of 23. This takes us to (x _ 1)(x _ 24), where the blanks will have opposite signs. The sign of b is negative, so the negative goes to the bigger factor:
(x _ 1) (x - 24). Then put the positive in the last blank (x+1)(x-24)
Back to the question:
Make factor pair lists of c. To get b, simply add the factors if c is positive, subtract them if c is negative. You can make b either sign in either case.