Why do we 'complete the square'?

The quadratic formula is really useful, but its derivation is confusing to many. Like what is the point of "completing the square" anyway? In this video, I show how "completing the square" has a geometric interpretation--you are actually making the areas of two squares equal to each other!

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"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 5/5 stars on 3 reviews)

[MindYourDecisions]

Professor Po-Shen Loh devised a different way to solve quadratic equations that could be helpful to many students.

[mandolinic]

I've been using that formula for over 40 years and until now I've never wondered where it came from. Thanks for the enlightenment.

[Frances]

This video should be presented in every high school and middle school algebra class.

[f.demascio]

I really enjoyed this video. Years ago, I was in college, majoring in Mathematics, with the goal of becoming a math teacher. My ultimate goal was going to be the teacher who spotted the "hole" in a student's understanding of mathematics, transforming them from a math-phobic student to a math-master student (as was done for me by a wonderful teacher in high school.) Since the field of math is cumulative, knowledge wise, any area not completely understood by a student in the past, will ultimately become an abyss in the future.
Sadly, a TBI ended my progress in college, but with the help of YouTubers like you, I am re-discovering my love of mathematics. Who knows? Maybe I can tutor some students one day.
Thanks!

[ksumar]

Gosh why didn't the school teacher explain this theorem in this illustrative manner. God bless you Sir 🙋☺👍.

[conradgarcia]

I never liked mathematics when I was in my younger age because most of my teachers presented it arbitrarily and non-chalantly, devoid of passion and wonder. Now, in my late 30's, I'm hooked on it because of great math channels like you. Can you do a series on millenium prize problems? Also: Hilbert's problems, Continuum Hypothesis. Thank you!

[kutlesss]

Wow. 'Completing the square'. Well, I'll be damned, that's exactly what we're doing.

[SabiqHaque]

Understanding the basics is a critical part of learning; unfortunately, not many people know that. We need way more people like you that not only understand that statement, but also willing to teach it. Keep doing what you are doing!!!!

[nakamakai]

At the time (1973), I thought my 8th grade Advanced Math and my 9th grade Algebra teachers did a fabulous job of presenting the quadratic. I'm willing to concede that this is better. Very well done. 5 stars.

[ianjohns]

why the fuck have i never encountered a teacher who has taught this?

[DxRzYT]

5:10 you could've explained this as when square rooting both sides you will go from (x+1)²=16, to x+1 = [+/-]4 because when square rooting, you get a negative and postive value. I just thought giving the length a negative value is quite confusing as that's quite impossible..

[Jukka]

I love how simple he makes it all look, by using a calm voice and great representations. And Desert Cactus you are right, this should be shown in every high school and middle school algebra class. I teach adult ed, and they need to understand algebra to get their HiSet or prepare for the test to go into college, and this is yet another excellent video from him.

[gregoryharlston]

I was struggling to teach some of my students the quadratic formula and I started thinking about it in similar terms such as this (minus the part about negative lengths). I knew some of my students might find this confusing. However, I took a chance and tried explaining the quadratic formula to my students in this way. For several of my students, I could see light bulbs going off--they got it! However, for many of my students, this only confused them further. I like your video. You explained it much better than I think I did (the colorful virtual manipulatives you used are extremely useful), so I think I'll be showing my students this in the future. I am a new math teacher and I try so hard to teach my students an intuitive way to think about math, like this for example. But sometimes, it seems that I only confuse them more. Also, if you've never been taught to think about math intuitively, then thinking intuitively requires a different orientation--kind of an undoing of just memorizing stuff to get answers. Keep up the great, great work!

[AlpieInternet]

Everyone made fun of me using completing the square in 12th grade, whilst they were using the quadratic formula - they said I was wasting my time. I believed it was a lot more elegant. So I decided to prove to them that it was the same thing during one of my classes near the end of the year... took me forever - but I eventually got it. IF ONLY I knew this. SUCH an elegant proof - great video explaining it!

[HollywoodF]

I was just realizing that I have followed every explanation you’ve every given on your channel with a minimum of effort. That’s what makes you a great teacher, and that explains why I keep coming back.

[worldhistory]

This is one of the best videos on youtube. I am in grade eight and had no problem keeping up due to your amazing visuals and your simple explanation of the concepts. I look forward to using the quadratic formula with the understanding of how it came to be!

[AgentMidnight]

I can think of a lot of people who could've benefited from this means of demonstration; much more straightforward and powerful than just forcing them to memorize the quadratic formula.

[sculpter]

that makes a whole lot more sense now. thank you

[olineolineoline]

This was really terrific. You did such a good job bringing us along and the visuals were super clear. Thanks for doing this!

[damiannagel]

Ok this is awesome. I've read on school books that the proof of the quadratic formula was too hard, but this is very simple and ties in with the concept of completing the squares. And you can even show it with pictures, for people that easily loses track of numbers.