How to factor $-x^2 + x -10$?

Question

This question has been killing me for hours. None of the factors of $10$ (because $(-10)(-1) = 10$) add up to $1$. So how do you do this question?

Answer 1

You are quite right, this is a case where it's not factorable. You could try using what is called the AC method which is when you multiply both the first term by the third and then find factors that add/subtract accordingly to the middle term (known as the b term).

I won't list out all the factors, but thinking along the lines of this is sometimes helpful:

$-x^2+x-10$

$1*10 = 10$ (and by your question, you have been looking for factors which is on the right track).

So, in general, we are looking for factors of $10$ that give us the middle term which is $1$. However, you actually realized that no matter what you were doing, you couldn't factor it any further. That's usually a sign it cannot be factored any further and that's ok. There are some equations/statements that cannot be simplified further. Mathematicians call this the factored form.

You could go ahead and rewrite the trinomial you were given as:

$-(x^2-x+10)$ (just to clean up the negative in the original statement, but remember it never disappears).

So technically yes, you could factor out a $-1$ but that's about all you can do in this problem.