Complete the square like a damn sage.

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:

$ax^2+bx+c=0$

You can do some acrobatic shit and make it like this:

$a(x+\dfrac{b}{2a})^2+c-\dfrac{b^2}{4a}=0$

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:

$x^2+bx$

We could just convert that to $x(x+b)$ , but let’s be a little more fucking original, shall we? Think about this shit geometrically. We could represent $x^2\, \mbox{and}\,bx$ like this:

complete1 — With me? The big square is $x^2$ and the rectangle is $bx$ .

With me? The big square is $x^2$ and the rectangle is $bx$ .

complete2 — Split that shit up. Now $b$ is split into two segments of $\frac{b}{2}$

Split that shit up. Now $b$ is split into two segments of $\frac{b}{2}$

Rearrange a little bit. Now there’s just a little square in the corner missing from the big square…

complete4 — The little square is $(\frac{b}{2})^2$ . Obviously.

See what we did there, chief? We took $x^2+bx$ , sliced and diced that son of a bitch and found that the only thing missing from making the whole thing add up to $(x+\frac{b}{2})^2$ is $(\frac{b}{2})^2$ . So then…

$x^2+bx+(\dfrac{b}{2})^2=(x+\dfrac{b}{2})^2$

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:

$x^2+bx+c=0$

Now, hopefully your ass will know what to do now. We have $x^2+bx$ . Looks familiar. Complete that shit. We have to add $(\frac{b}{2})^2$ . Don’t get too hasty though, boss. You have to add it to both sides. Keep that shit balanced.

$[x^2+bx+(\dfrac{b}{2})^2]+c=(\dfrac{b}{2})^2$

Square completed. Now factor that business.

$(x+\dfrac{b}{2})^2+c=(\dfrac{b}{2})^2$

Bring it all to the left and make it equal to zero like it was before.

$(x+\dfrac{b}{2})^2+c-(\dfrac{b}{2})^2=0$

$(x+\dfrac{b}{2})^2+c-\dfrac{b^2}{4}=0$

One more thing. $(x^2+bx+b^2)$ is fucking peanuts to factor. $(ax^2+bx+b^2)$ though, not so much. Not a problem, y’all. Divide all that shit by $a$ . Then you get $x^2+\dfrac{b}{a}x+\dfrac{b^2}{a}$ . Problem solved.

Example time, bitches.

Por ejemplo

Solve $2x^2+4x+3=0$

Step 1: get that two out of there.

$x^2+2x+\dfrac{3}{2}=0$

Alright, now let it rip, homie. In this case, looks like $b=2$ and $c=\frac{3}{2}$ .

$[x^2+2x+(\dfrac{2}{2})^2]+\dfrac{3}{2}=(\dfrac{2}{2})^2$

SQUARE FUCKING COMPLETED

Keep on keeping on.

$[x^2+2x+(\dfrac{2}{2})^2]+\dfrac{3}{2}=1$

$(x^2+2x+1)=-\dfrac{1}{2}$

$(x+1)^2=-\dfrac{1}{2}$

$(x+1)=\sqrt{-\dfrac{1}{2}}$

$x=\sqrt{-\dfrac{1}{2}}-1$

$x=i\sqrt{\dfrac{1}{2}}-1$

Damn… finally. Got our answers:

$x=i\sqrt{\dfrac{1}{2}}-1\,\,\mbox{or}\,\,x=-i\sqrt{\dfrac{1}{2}}-1$

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.

Comments

Arthur Philip Dent

October 24, 2013 at 6:01 am

That’s actually really cool–visualizing the process with squares and rectangles like that. It makes it a lot faster and easier to grasp what’s going on.

- Grownass Man
  
  October 24, 2013 at 3:53 pm
  
  Right? Too many damn people just memorize this shit – hardly anyone has a clue as to what they’re actually doing and why it works.
  
Alissa

November 11, 2014 at 7:58 pm

Keep posting! This stuff is so good!!! Love your site!

Jonathan Dunce

April 9, 2015 at 11:59 am

After reading this I still have no idea what this is. The explanation doesn’t tell me why we do this. I guess I’m stupid, but I understand everything leading up to this so far and I’ve read maybe 6 or 7 “no-nonsense” explanations and I still have no idea what’s being done or why or in the case of some of the calculation why the calculation works the way it does. What is the ultimate point of completing the square?

- Grownass Man
  
  February 4, 2016 at 9:59 am
  
  Completing the square is just another tool that we can use to factor equations. In particular, equations that can’t be elegantly factored some other slick way. The ultimate purpose of factoring is finding solutions for equations. That’s what it’s all about, yo! Finding solutions!
  
  - Sam
    
    April 22, 2018 at 7:04 pm
    
    Comes in HELLA handy in advanced calculus i use your stuff for my engineering calc 3 and it makes the concepts make more sense
    
Hannah

December 6, 2016 at 3:13 pm

I’m a really visual learner and my teacher has tried to explain this to me so many times, and it never clicked. After reading this, which only took me ~5 minutes, it seems super easy. Thank you!!!

Robert

December 7, 2016 at 5:44 am

What if b is a negative?

- Grownass Man
  
  December 7, 2016 at 11:13 am
  
  Don’t make a damn difference! Just use it like normal. Same basic arithmetic applies, yo.
  
alexwebbbAlex

May 16, 2017 at 7:24 pm

I had a similar reaction to Jonathan Dunce up there. After doing a little bit of research, I learned that equations which are square (that is, a square root of the equation can be found) are easier to factor. I think before I learned that, I just thought that it was just some sort of strange word game, and didn’t understand the distinction between an equation in which the square root can be found and an equation in which the square root cannot be found (at least until the square is completed). Until I understood that distinction, I couldn’t understand why the word “square” was being used. Anyway, thanks for the tutorial.

John Doe

February 14, 2019 at 6:29 pm

Thank you so much because my teacher can’t teach and nor can khan academy.

mr.

May 25, 2019 at 5:00 am

you’re all over the place on this one for me.
after the w t fuck nobodies did you switch to another example?
i need complete clarity and a crumb trail when learning something new.

mars

June 26, 2020 at 3:37 pm

this website is the only thing keeping me from failing dude thanks for this shit