Benjamin A. answered • 04/26/19
Ben's Math/Stats Tutoring
The first thing you should do when you are factoring is look at the "c" term, or the term that is at the very end in the quadratic ax^2+bx+c. When noting the "c" term, you should look at the sign of the term, whether it is positive or negative. This "c" term happens to be -1. The "c" term is found by multiplying the final two components of your two binomials, (when I say two binomials I mean something that looks like the following (e+f)(g+h), I call this a double bubble). When multiplying the two final terms of the two binomials, consider how you could possibly end up with -1. The only way to end up with a negative when multiplying is by multiplying a positive number and a negative number. Therefore, the two numbers in our binomial are opposite signs. The only way to multiply to get a 1 is by having a 1 multiplied by 1. Therefore we know the two terms in these binomials have to be -1 and 1.
Now consider the first term in the quadratic. It happens to be x^2 with no number in front. Therefore the only way to multiply the first terms to get an x^2 is by multiplying x times x.
Fitting this into our two binomials we have (x-1)(x+1). However this is not correct!!!! Because when you multiply each term of the binomial you would have x^2 +1x -1x -1. Then, the middle terms would cancel out.
So we conclude that it is not possible to factor this quadratic. The only way to solve it is by the quadratic equation, and if you did so you would have complex roots. We can be sure this is the case because the final "c" term has no other factors than -1 and 1, however, when we put this into our binomial it doesn't factor correctly. In other words, this binomial is prime.
