Alright kids, I'm coming at you with something we all learned in algebra - only this time with a fucking explanation that makes some damn sense. I'm talking some completing the square shit, y'all.
For those of you that don't know, completing the square is when you have some polynomial like this:
You can do some acrobatic shit and make it like this:
Now right about here is when most people say "W T FUCK" and give up like a bunch of fucking nobodies. Don't be a nobody, asshole. Learn something.
Complete the square.
Now, this trick is some slick-ass factoring kung fu. Consider this expression:
We could just convert that to , but let's be a little more fucking original, shall we? Think about this shit geometrically. We could represent like this:
See what we did there, chief? We took , sliced and diced that son of a bitch and found that the only thing missing from making the whole thing add up to is . So then...
Done and done. That square is fucking completed. Makes a bit more sense drawn out rather than just memorizing ambiguous-ass formulas, right?
Now, factor some shit.
Alright, so now we can use this for factoring. Consider this generic-ass equation:
Now, hopefully your ass will know what to do now. We have . Looks familiar. Complete that shit. We have to add . Don't get too hasty though, boss. You have to add it to both sides. Keep that shit balanced.
Square completed. Now factor that business.
Bring it all to the left and make it equal to zero like it was before.
One more thing. is fucking peanuts to factor. though, not so much. Not a problem, y'all. Divide all that shit by . Then you get . Problem solved.
Example time, bitches.
Step 1: get that two out of there.
Alright, now let it rip, homie. In this case, looks like and .
SQUARE FUCKING COMPLETED
Keep on keeping on.
Damn... finally. Got our answers:
So there you have it... solving by completing the square. Not exactly the most elegant shit but it gets the job done. There are plenty of times when it really is your only option, so keep it in mind. It's also, by the way, the first step in deriving the fucking unholy mess that is the Quadratic Formula.
Don't just memorize it. Understand it, ya roody-poo.