Factoring tips for 'find two numbers...' problems

[gopher]

Having trouble with this sort of factoring exercise:

Find two numbers that multiply together to equal one number and add to equal another number.

So far, I'm just randomly guessing numbers. Is there a easy formula, method, or tip that can make the process faster?

 

[galactus]

You don't have to randomly guess until you find them. This is mathematics, afterall.

Let's take \(\displaystyle \L\\x^{2}-2x-15=0\), for instance.

You want it in the form \(\displaystyle \L\\(x+a)(x+b)=0\)

You want to factor, so you want two numbers which when added equal -2 and when mutliplied equal -15.

\(\displaystyle \L\\a+b=-2\)
\(\displaystyle \L\\ab=-15\)

Solve this itty-bitty system and you find that a=3 or -5 and b=3 or -5.

\(\displaystyle \L\\(x+3)(x-5) \;\ or \;\ (x-5)(x+3)\). same thing.

Therefore, the solutions are x=-3 and x=5

After you find the numbers you need, you can also set it up like this:

\(\displaystyle \L\\x^{2}+3x-5x-15\)

\(\displaystyle \L\\(x^{2}+3x)-(5x+15)\)

Factor out something common:

\(\displaystyle \L\\x(x+3)-5(x+3)\)

Note what's in the parentheses is the same. Must be to work right.

\(\displaystyle \L\\(x+3)(x-5)\)