Completing the square (step by step instruction & visual illustration)

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#math #algebra #mathbasics

[noahblack]

In 7th grade I moved up a level in math. I skipped the second half of Pre-Algebra and the first half of Algebra 1. Went all the way to multivariable calculus in college without ever understanding what people meant by "complete the square". Thanks for finally teaching it to me in a way that I can already tell will stick.

[TGears]

The visual aspect of this was something I’ve never seen. I’ve always loved math, so much so that I finished all math electives in my school district before I was a senior in high school, I took enough courses in engineering school to almost get a major in math too, and I have never ever once seen a visual for completing the square. It’s so obvious when you point it out but I haven’t ever had my brain process it that way.

Thanks for the great and concise video!!

[a.tsuruya]

I love how this method pretty much tells us how the quadratic formula works.

[jannegrey]

Thanks! As a person who uses quadratic formula only, I was always surprised how fast people can factor such equations - and factorization that you've shown here (and explained in another video), kind of explains it. I knew about completing the square, but I never learned how to do it. Even though I watch mostly your calculus videos, honestly the "basics" channel is most educational. Because even though I like maths and I'm on higher level than such basic maths - it shows how often I have gaps in knowledge that I should have learned 20+ years ago.

Question: There is also a "version" of quadratic equation (IDK how it is called in English) that uses values p and q IIRC. And it is different "notation" of the same equation. I honestly forgot what it even represents (it might be factorization method maybe?), I just do remember it exists. Do you have a video on that? I remember it being "less useful" and "harder to derive" than quadratic formula ox ax^2+bx+c, but 1 or 2 things were easier with it. I completely don't remember what it was used for though.

[Mediumcoffee]

I really appreciate you showing the geometry part. If I ever knew it, I had totally forgotten it.

[bprpmathbasics]

Where did the quadratic formula come from?

[geralynpinto]

Great! I only knew how to complete the square algebraically. But I did not know about the geometric representation. Thanks so much! Everyday I learn something new.

[kahrkunne]

This is cool, I've always heard people talk about "completing the square" online but never knew what it meant, I don't think I was ever taught it here in Europe.

[omari]

This is one of the most perfect explanations of something I’ve heard in a long time. Maybe folks have been full of shit lately, but you’re not sir. You’re a wonderful teacher. Teachers are awesome. See?

[AllanPoeLover]

但是我突然想不通
如果幾何意義上成立
那 根號13 < 4
所以 X = -4 +- 根號13 兩個根都是負的
這個時候 X 等於負的 在這個正方形上 的意義是甚麼 ?
長度有 負 的 嗎?

[AzureKyle]

I actually did solve with the quadratic formula, just for fun, and ended up with x=(-8+/-sqrt(52))/2. The square root could be simplified to +/-2*sqrt(13), which with it and the -8 being divided by 2, left me with the same answer of x=-4+/-sqrt(13)

[theobjectivethinker]

So the square root of 13 is approx 3.6055513, how does it all add up if one was to input that number.

[pulsar]

This method never got taught when i was at school (1980's).

[glenndeprey]

Why do other teachers make it so much harder then u do haha

[JubeiKibagamiFez]

Is this practical math or theoretical? I keep looking at the original equation and trying to solve it without all these gymnastics. So, X²+8X+3=0.... Without PEMDAS, X=0 and the 3 goes first, then it would equal 0. So, 3+X²+8X=0

[Frat]

It's probably the worst explanation of completing the square I've seen.
So, for everybody who actually want to understand why this works:

We have a formula, that says that (a+b)^2 = a^2 + 2ab + b^2
So, when we have equation like x^2 + cx + d, where c and d are some constants, we want to turn that thing into (a+b)^2
To do that, we already have our a^2, where a is x, and we also have some cx which we can see as 2ab. Now we need the b^2.
For that we need to know the b. So, if a is x, 2ab is 2b*x. our cx is 2b*x, so 2b = c, b = c/2.

So our square, that will be (x + c/2)^2 = x^2 + cx + c^2.
But we have x^2 + cx + d instead. So we need to write d as: (c/2)^2 + (d - (c/2)^2)
Since (d - (c/2)^2) is some constant, let's say it's z
So now we have (x+c/2)^2 + z
We can subtract z on both sides of equation now, so if we had (x+c/2)^2 + z = 0 we'll have:
(x+c/2)^2 = -z
Then we'll just have square roots on both sides, so x+c/2 = +-sqrt(-z).
And x = +-sqrt(-z) - c/2.
Note that z itself might be positive or negative.

Same thing is possible for x^2 -cx + d. But this time it'd be (a-b)^2 formula, which is a^2 - 2ab + b^2.