Deriving the Quadratic Formula 

Why do math teachers in high school never show how to prove stuff. They just say here's the formula now DO MATH. Takes out all the fun out of math. No wonder so little people in my class like it.

I can't believe I was never taught this. Thank you, puts some meaning into the formula.

Omg thank you! I started giggling when it all worked out

Great explanation! In high school I remember deriving this but that's been over 50 years and now I'm teaching my grandchild so I wanted some reminders.

THANK YOU THANK YOU THANK YOU A MILLION TIMES!!! THIS WILL BE ON MY MATH FINAL EXAM TOMORROW!!!!! I CANNOT THANK YOU ENOUGH!!!!!!!

@azk2020 well, often times people want to know why something is true. showing why something is true (some formula) is usually much more difficult than showing how it is used. i think just about everyone has seen the quadratic formula by high school, so i think it is rewarding to know where it comes from : )

aww thanks : ) glad i am able to help a bit

Great vid! My math teacher in Algebra 1 my freshman (1989) year had us practice completing the square every day for weeks. After doing the repetitive task several times, I knew there had to be a formulaic way to get the answer. So one evening I went home and figured it out exactly as you showed here. I went to school early the next day to wait for my teacher to get there to show him what I discovered. I was very excited because I thought I was going to get my name on a function that hadn't been discovered yet. He was extremely impressed that I figured it out, and then smiled and had me flip my algebra book ahead a couple chapters to the quadratic formula. While he was excited for me, he totally let the wind out of my sails. Three years later when I was a senior, he wrote an incredible letter of recommendation for me when applying to a university to study Actuarial Science. I got accepted with a very nice scholarship to boot!

Thank you so so much for making this video! I was really confused when my math teacher explained this concept today, but your video has really helped me clear things up.

u are a very good tutor...keep it up

Deriving is the best lesson for me ,I love to deriving in mathematic.

The question suddenly came into my mind as to where the quadratic equation driven from, just go to you tube and find you there. Awesome! Thank you.

well, completing the square is basic high school algebra... i think most any decent hs algebra student could follow this

3:24. I keep forgetting what I'm actually needing to add back in! I kept adding b^2/4a^2 to the right side. All the other examples online I found took the a across as the first step so I wasn't able to discover my problem. Thanks! This often catches me during completing the square - neglecting to pay attention to the constant outside the parentheses when deciding what needs added back in.

Hey, I really want to let you know how great this video was. Maybe I am a weird person for being like this, but I can't bring myself to use stuff in any sort of subject until I know exactly why it is that way so you have definitely helped me understand how the quadratic formula is found and why it makes senses to use it for problems. :D

I remember how cool it was in 7th grade to go though this in my math class. This brought back such a great memory.

Genuinely thank you so so so much. I was stressing over this earlier today and now I have no worries about it and I fully understand the concepts. Thank you again.

I taught my students how to do this by completing the square on 3X^2 + 6x - 12 = 0 . They are proficient at completing the square with numbers at a, b, and c terms. I then put the above mention equation on one side of the white board, and we followed the steps onto ax^2 + bx + c = 0. They were able to quickly understand each step due to their familiarity with completing the square. These are 9th grade freshmen.

I am 30 now and this question popped up in my mind while solving 2nd degree polynomial equation without calculator that how this formula arrived. In school and college i didn't have enough time to question my self how this formula arrived. Now I m an electrical engineer in power generation company. Thank you sir that was pretty impressive 👍😊.

I'm an 8th grader this year and we have to take a test tomorrow over Ax^2+Bx+Cx and finding the quadratic formula. I am SOOOOO grateful for this video!!!😄😄😄

have a test on this tomorrow and need to have this memorized... wish me luck

my teacher is trying to make me solve for this during middle school while everyone else here is in high school

I missed school when my teacher explained this, so I'm really glad I found this video. Especially since it's gonna be on a test.

After 20+ years of wondering, I finally got focused enough to actually look this thing up. As someone who tries not to take things for granted, this bothered me for decades. But it would only be flashes I would end up ignoring. Today was the day I said: NO MORE! It's fascinating how it all fits perfectly after the "perfect square" gymnastics. Like, the same denominators on both sides made it smooth. Math is magical, too bad I was kind of an idiot in youth and gave up on education

Damn I'm commented on 2019 here lmao. Also Jesus i really suck at math i just wanna say thanks for putting an effort to this video really helps me a lot.

Thank-you so much for this. It was set as homework and I made so many 'silly mistakes' which meant before I watched the video I had spent hours trying to complete it myself. You have 'showed me the light'!

Thanks so much! I've been wondering for over 15 years how this was done but never set out to find out, until now with your wonderful video! ;)

From class 10 cbse Really helped a lot sir Great explanation From India 🇮🇳

just make sure you understand the process, then be careful with all the fractions! good luck!

Thanks a lot. I've got a quiz on this tomorrow, and this is just what I needed.

Thanks so much, Patrick! I'm shamed to say that I never knew that, yet I have a math degree. I never had a clue about how to isolate the x . How on earth could anyone criticize your great presentation? They could look elsewhere if they didn't like yours. I'm a southpaw just like you, and our hands tend to cover what we are writing. I just wanted to see every step shown the second you were writing it. Now I can teach this to my advanced students. Again, thanks so much!!!

Thanks for making this video! I have a quiz on this today so this really helped me out!

Absolutely delicious! A very smooth and easy to understand approach.

@PSilysvmftbomhx3 yes, i try to make all my videos like this: effective and efficient

Thanks for reminding me. I've forgotten that we had to do that in algebra. Very useful.

I don't know if anyone has said this, but deriving the quadratic formula is a CSET question. Thanks for explaining it!

Bro thank you so much I'm taking a math test at home right now and this is question 7 of it

The derivative is defined as the limit as h approaches 0, of f(x+h) - f(x) all divided by x+h-x. It is the slope formula except for a point that is infinitesimally close the value of x. If you set h to zero and solve the resulting limit based on f(x). This will then give you the formula for the slope at any point on a curve! It just turns out that this formula always gives the answer of reduce the exponent of n by 1, then multiply by the old exponent. Hope this helps but I'm in grade 11.

There is even more simple intuitive proof. Middle part is sum and last part is product. So S=b/a, P=c/a now you have sum and product, you can calculate both routes. Say your routes are R1 and R2. Do not substitute R1 = S- R2 you will again end up in quadratic equation. Instead, use this formula. (R1-R2)^2 = R1^2 + R2^2 - 2*R1*R2 = (R1 + R2)^2 - 4* R1*R2 = S^2 - 4P Square of Difference = square of Sum - 4times Product Difference D= sqrt ( S^2 - 4*P ) Now we have difference and sum so solution is R1 = (S+ D)/2 R2 = (S-D)/2 Substitute S and D, you get answer. So basically you know Sum and Product, so calculate Difference and u are done (a+b)^2= a^2 + b^2 + 2ab (a-b)^2 = a^2 + b^2 - 2ab R1

always happy to help out a super star

@iamcool1908 ha, most people are nice, but it is the internet, you can not please everyone all the time! glad you like it though!

That was really cool! I always thought this fella just went,”I’m gonna find something that works for all quadratic equations” and ta da he does. At least that was what I thought

this one is a little bit simpler ax²+bx+c=0 shift c to right side ax²+bx=-c divide both sides by a x²+bx/a=-c/a or x²+(b/a)x=-c/a Add (1/2*coefficient of x)² on both sides x²+bx/a+(b/2a)²=(b/2a)²-c/a x²+2(x)(b/2a)+(b/2a)²=b²/4a²-c/a [(x)+(b/2a)]²=(b²-4ac)/4a² take square root on both sides x+b/2a=±sqrt(b²-4ac)/2a shift b/2a x=-b/2a±sqrt(b²-4ac)/2a OR x=[-b±sqrt(b²-4ac)]/2a hope this clarifies

Awesome vid man! I'm starting Calculus 3 in the spring, and that equation just always seems to keep coming back! lol Now I finally know where that beastly equation came from. Thanks for showin it!

my algebra teach came up with a genius way to memorize the formula. here it is: the negative boy (-b) was confused (+ or -) if he wanted to go to a radical party ( the square root box). the boy felt like a square ( b squared) because he missed out on four Awesome Chicks ( -4AC) the party ended at two a.m. (2A). if you put that in the order of the formula you will see it corresponds

Thanks for this helps in our school the teachers never teach prove (examination point of view) but I really am thankful to you for this

@VIR092 i like it too : )

no problem!!

Gotta say my teacher is god for actually teaching me how to do this. But honestly a great video to review

I'm always forgetting the exact formula, and it is great to be able to simply derive it.Of course, I had also forgotten how to derive it... oh yeah, completing the square. The cobwebs begin to clear. I appreciate everything you do, Patrick. TFS!

Omg thank youuuuuu! I was so confused in this concept! Thank you very much for explaining this! My idiotic book just wrote it directly instead of explaining the points 😭😭😭 Finally I understand :'D

Thank you sir 🙏my age is 13 years and u am watching it after 13 years whole concept clear .

excellent! just complete the square and be careful with all the letters floating around

@geraoma you are very welcome : )

It's so fascinating how almost everyone takes this for granted and never thinks about what high level math went behind the formula they use on a day-to-day basis. Liked and subscribed, keep up the amazing content.

Why does a math video over a decade old teach me more than anything they taught in math class...

that‘s why i love math，math is beautiful

Thanks alot for this, your probably the best teacher teacher on youtube and the you would own the math teachers at my school!

Very well explained i love watching derivations

Thanks man. I needed to know this for my Geo class

Thanks very much. Was looking for something like this all over

Bruh this was released when I was exactly 3 and a month old. Now I’m in my freshman year of high school. How time flies😩

Thanks for the cool demonstration of how the quadratic equation came to be. I saw it in class today and was dumbstruck at its emergence. Math is fun!

I love this video. I have to memorize how to do this and your video is very well explaining and makes it a lot easier to remember. Thank you.

WOW! This is gonna sound like a cheesy testimonial thing, but until I saw this video, I was TOTALLY stumped as to how the constants '4' and '2' got into the quadratic formula. Thank you, Patrick, for making this concept so understandable via RU-vid video. As to the person who said Patrick explained it too fast, you can always stop the video, use the round video "counter" to go back to the point where you get stuck, and look at the equation, and think it through for as long as you need to.

MAAAAANNNN YOU JUST SAVED MY LIFEEEE. THANK YOUUUUUUUUUUUUU

Excellent explanation. Patient and thorough. Understood it first time round. Thank you.

thank you very muCH! five points bonus on a big test if we could derive the quadratic formula. and believe me those points were much neeeded! thanks again!

thank you for posting this. I wish teachers would work out how to derive the formula more, it really helps you know what you're doing

Thanks @patrickJMT this´s a really helpful staff that many teachers miss. Could you make some more videos deriving quadratic formulas? ... like the discriminant, intercept, or the vertex form. It happens that many times they teach you a formula, but you don`t really understand the how did it came, this is a great way to understand math better (Y). Thanks you again.

thanks for making this video...its helps a lot for cause as far as I know deriving the quadratic formula is actually, one of the question on the Engr board exam.

This is a life saver, this is in my test tomorrow tysm!!!

Great explanation and some wonderful moves on the maths. I ve been trying to derive this for myself on and of for over fifty years ; so you can imagine how helpful this was - I can do it and I understand it (thanks)^2

Dayumnnn! This is legit cool explanation! THANKS MAN!

Divide by a after writing the second line, it makes the whole thing a lot simpler than waiting till later to divide by a. We can do this since a is not 0.

Finally learning to derive the formula. Teachers are terrible at this. They expect you to memorize the formula. I bet some can't even derive it. If you learned how to derive it kids wouldn't need to memorize it.

I believe most of the guys could not get-----multipy 4a top bottom @5 12 so this is to force lcm to be 4a>2 when c over a is multiplied. The same can be done with digits in fractions.

EXCELLENT Teaching..... Hallmark!....wow!..... Thanking you Sir. .From India. But we Indians feel difficulty in understanding slung Anerican..

Very nice and high quality video considering it was made nearly 12 years ago lol.

Cheers mate, your a lot more better then my Scotish teacher, (i am currently doing year 11extension)

Thank you bro...This video is very useful and easy to me..so again thank you☺☺

I was taught a method where you started by dividing out the coefficient a, we didn't use factoring except in one or two steps but great tutorial :3

im gonna hv a test next week and this video rly helped, it was actually kinda entertaining (even tho maths is d most boring subjects out there)! now i'll go through all ur other videos ^_^

@patrickJMT. I am confused as to how you get 2a after taking square root of 4a^2. Doesn't it have to be plus or minus 2a? Since you square rooted the a^2?!

THis was on of the question s on my pre-calc project

Thank you so much! I have a test tomorrow and I KNOW this is a question! THANKS!!

on the first step wouldnt it be a ( x^2 + b/a + c/a) = 0 ? because if u leave out the c and move it to the right side of the equation (like you did) the formula wouldn't work if A was equal to 0 although we would get the correct answer: the quadratic formula at the end. just a question i had in my mind

It's fascinating, but I can't replicate this. I guess maybe I need to revisit Algebra 1 and 2.

Man, that was awesome! A bit like learning about the creation of the Pythagorean theorem in-so-far-as it's applying known information to make a brand new method that is much simpler and faster to use that can be applied to all equations of that subject area. GREAT EXPLANATION PATRICK!!!

sir . when u factorise taking a out and ure leftwith x^2 + b / a multiplied by x in the next step where does that x disappear ?

great demonstration man.

yes you make me happy. and I'm not gonna lie, that was confusing, trying to derive stuff in trig right now. Its hard but you explain thing so well. keep going!

Thank you for this very helpful video. I have a BEng in electronic engineering, as well as twenty years work experience in IT, working for many top organisations, mostly investment banks but also Microsoft and other 'blue chips'. However, found very poor mathematics teaching, in particular, decent work-throughs and explanations like this. Well done and keep up the good work! Will contribute something next time I have a little spare cash (-;

when you completed the square why did you add b^2/4a^2 to the left side but on the right side the 4a isn't squared? shouldn't they be the same?

Man, that was really cool. A bit like learning about the Pythagorean theorem in-so-far-as it's a really interesting way of applying what you already have to come up with a brand new method of working equations in a much simpler and faster way. Great explanation Patrick!

I love deriving this for people. It makes you seem like a total genius, when the only real secret is completing the square of a general quadratic equation.

Love it. Very concise and easy to follow.

Help im confused Why did you take 1/2 of whatever the number infront of x??

thanks for the useful comment

In 8th grade I took Algebra 1, so all the kids in my class always wondered stuff like this, so we would go through it