Is There a Faster Way to Factor Than Just Guessing Factors?

[Tyrion101]

One of my quadratic formula problems required me to factor so that the product of 2050s factors added to -23, that may not be exact, but I basically asked for a new problem since it was just practice, after an hour of getting close but never the right answer, and I was wondering, other than just using the computer program I wrote (which would likely be cheating) is there anything that gets me an answer faster than just guessing factors?

 

[mfb]

So you were supposed to find numbers a,b such that their product is 2050 and their sum is -23? That is not possible (at least not with real numbers).

Anyway, I would do it like that:
As a+b=-23, we can write b=-23-a.
Plug this into a*b=2050 and you get a regular quadratic equation. With the numbers in your post, it does not have solutions, but with other numbers it has.

 

[verty]

You had something like x^2 - 23x + 2050?

2050
2 - 1025
5 - 205
5 - 41
41 - 0

2050 is 2.5^2.41

Knowing the small primes helps. For example, I knew that 41 is prime. But also, there is an easy way to check. If a number n is composite, it has a prime factor ≤ $\sqrt{n}$. So with 41, if it is composite, it has a prime factor < 7, so check 2,3,5, none divides it, so it is prime.