How do you factor quadratic x2−x−56?

2 Answers
Alan P.
Mar 27, 2015

Solve x2−x−56=0
using the formula
x0=−b±√b2−4ac2a

to get solutions
x0=a and x0=b
(in this particular case x0=8 and x0=−7)

Since the original quadratic is equal to zero when x is equal to either of the x0 values
(x−a) and (x−b) are factors of the original quadratic.

Using a=8 and b=−7
(x−8)(x+7)=x2−x−56
and the factorization is complete.

Jim H
Mar 27, 2015

To factor x2−x−56 by trial and error, use the following.

(This takes longer to explain than it does to do.)

We're looking for
(x+b)(x+d)=x2+(d+b)x+(bd)=x2−x−56

the products used to get 56 (bd) using whole numbers are
1⋅56
2⋅28

4⋅14 (3,5,6 are not factors of 56)

7⋅8 (the next number, 8 has already appeared on the right, so we're done)

We want −56, so we need b and d to have opposite signs. (One is positive and the other negative) they need to add up to −1 because we want a −x=−1x in the middle. +(d+b)x=−1x, so d+b=−1.

Here's the list again:
1⋅56
2⋅28
4⋅14
7⋅8
If one is positive and the other negative which pair do I want to get a sum of −1?

Looks like I want 7 and −8

Check to be sure

(x−8)(x+7)=x2+7x−8x−56=x2−x−56 Good, that works.

Note
If there is a number (other than 1) in front of the x2, we need

(ax+b)(cx+d)=(ac)x2+(ad+bc)x+(bd)

Related questions
See all questions in Factorization of Quadratic Expressions
Impact of this question
8808 views around the world
You can reuse this answer
Creative Commons License