Why Does the Quadratic Formula Work?

[@_crypt]

Our teacher taught us this wayy before even getting started with solving them, Im really grateful of that.

[@elithzy]

I'm jealous of you

[@thebetterbutter709]

"e" as a variable is diabolical

[@Prismate]

sacrilege

[@haygiya]

quick fix: 0:30 line in red intersects the line *Y=0*

[@striker8380]

Being pedantic, in this case, as the vertical axis is labelled f(x) it would be f(x) = O

[@IgnitionP]

@@striker8380 Being correct*, intersecting at x=0 is way different

[@montykemsley9479]

@@IgnitionPI think they meant that they were being pedantic by correcting y to f(x)

[@IgnitionP]

@@montykemsley9479 ah ok that makes sense, thank you

[@B5Scheuert]

​@@striker8380 I feel like being pedantic in a math context is quite understandable

[@shantom2340]

This is how math should be learnt instead of repetitive textbook problems. Amazing video

[@dosomestuff1949]

Both. Both conceptual understanding and practice are imperative to understanding math

[@hmm8464]

@@dosomestuff1949true

[@MotionisHere]

​@@dosomestuff1949ur right

[@n1xtertrixter241]

Thats literaly how i was taught the quadratic formula.

[@obeyy0urmaster]

You sound very uneducated

[@RPrice_OG]

More than 40 years after high school and never having to use it in real life I still remember it. That's crazy.

[@gregoryt1139]

I support the Engineering division of the power company. I would ask those who didn't present as "unique people," if they've ever said that in high school. Their responses always began with a chuckle.

[@CarlosSanchez-oe2ls]

I dont know anything about the program you use to do these videos but i think that using it is the best way to explain maths. Thank you very much.

[@risograph]

its called manim!

[@JellyfishJellyfish-bk7cr]

Probably community edition

[@champx5]

it's called manim. it was developed by 3b1b

[@usm1le]

this video is so clean in its format and i love the illustrations on screen. underrated

[@Arvinax-29]

Any more channels like this 😢😢. This is a goldmine of information.

[@tradersendeavors]

3Blue1Brown

[@ravindupathirage1884]

Yup.. 3blue1brown, the literal legend

[@JellyfishJellyfish-bk7cr]

+ Mathmagitian

[@JellyfishJellyfish-bk7cr]

Oh and numberphile of course

[@mra4167]

My teachers didn't teach me this but now i know how it is derived. Thankyou

[@AdamR-b7y]

Yeah, I mostly just used a factoring trick instead of using the formula when dealing wth quadratics.

[@lawrencema1985]

Nice video. This was how they introduced complex numbers to us at uni. You can see that you only get real roots if the thing under the square root is positive...

[@Pastpresentfuture280]

Thanks!

[@tanmaygupta1555]

Another proof can be Sum of roots= -b/2a Difference= √D/4a Solve to get roots

[@saisiddeshmamidoju1728]

did we not get these formulas from the main quadratic formula?

[@fallay2808]

@@saisiddeshmamidoju1728 nah its like viet's formula, let x1 and x2 be the solution of ax^2+bx+c=0 so we have a(x-x1)(x-x2)=0 , you expand that expression and u get those formula :)

[@tiesvinke8460]

3:50 wow thats such a nice moment when you realize the formula for the top of the function also comes from the quadratic formula

[@samkelly99]

thats pretty cool to know that theres some real logic behind the Q formula ! Thanks for shooting this video. Very cool

[@nimbus3957]

Videos like these really deepens my understanding on the topic. Hope we will keep seeing more videos like this. I subscribed!

[@fiaioiehizooiwhjd]

love this video, just wanted to let you know that the zeroes are where the function intersects y = 0, not x = 0 like you stated at 0:34

[@pi_squared2]

Thank you, and yes that's a good correction! I've put it in the description of the video.

[@raghavk-yb7rl]

Great video!!!! Really interesting, and well explained... btw 0:34 it's y=0

[@mikeriesco6174]

At 0:33 or so, you meant to say "...the line y=0"...

[@fanamatakecick97]

All you have to do to understand the quadratic formula is to not isolate x. x = [-b +/- sqrt(b^2 - 4ac)]/2a 2ax = -b +/- sqrt(b^2 - 4ac) 2ax + b = +/-sqrt(b^2 - 4ac) 4(ax)^2 + 4abx + b^2 = b^2 - 4ac 4(ax)^2 + 4abx = -4ac ax^2 + bx = -c ax^2 + bx + c = 0 Now, of course, this shows that you have to complete the square to actually derive the quadratic formula

[@aratros2766]

This so much made me want to learn more about abstract algebra

[@joaomonteiro1574]

That was ENLIGHTENING!! Thanks for this video with such good quality.

[@renesperb]

One could mention that this formula also holds if one or all the values a,b,c are complex numbers.

[@DarkAstro13]

my teacher always hates on me for using it instead of factoring... but it always works and is super simple to use without a calculator(often)

[@yeetingkelp8624]

Great explanation! Had me focused and ready to learn

[@ZachDrake5960]

0:34 - That line is the x-axis, or also known rather as the line *y* = 0.

[@akotoy]

Well made video, hopefully your chanel grows, maybe add music

[@razalasreficul6902]

Music would add nothing useful to the content and learning outcome. On the contrary, it may even be disturbing. It's great the way it is.

[@willburhans3018]

nah, keep the music out. Unconscious pandering.

[@paradox3667]

There is another very similar proof I like very much. It goes like: (x+p)^2-q=0 => x= -p +-sqrt(q), which is very simple to prove. But it can be written as x^2+2px + p^2 - q =0, which is similar to the standard notation ax^2+bx+c=0 or x^2 + b'x+c' = 0 (with b'=b/a, c'=c/a, this is just to make writing it easier but can be removed) If you group like terms you get 2p=b' and p^2-q=c'. solving for p = b'/2 and q = p^2-c' = (b'/2)^2-c' substituting in p and q in the solution we get x= -b'/2 +- sqrt(b'^2/4-c') = -b\2a +- sqrt(b^2/4a^2 - c/a) = -b/2a +- sqrt({b^2-4ac)/4a^2) = -b/2a +- sqrt(b^2-4ac)/2a = (-b+-sqrt(b&2-4ac)/2a I like this proof because you start with a clear end goal. "This form is easy to solve, so lets try to see if we can manipulate it to look like the more general case." For every step it's clear why its done.

[@SanadJaser]

I love the video extremely well made but your voice reminds me a lot like Iman ghadzhi lol it’s a good thing!! Keep it up :)

[@moonmissionpassagetototali1952]

You said at 0:36 that the function intersects the line x = 0, rather than the function intersects the line f(x) = 0, which is the x-axis. The line x = 0 is the y-axis. You should have said the function intersects the x-axis at the two points where f(x) = 0.

[@mathchannelforwatching-s4i]

I can't wait to see more from you

[@pi_squared2]

Thank you!

[@smabedi]

It might be interesting for you to know that the quadratic formula was actually invented by a Persian Muslim mathematician named al-Khwarizmi (in fact, the word algorithm also comes from the word al-Khwarizmi)

[@starpeep5769]

Tysm fun fact

[@arushjais]

He copied the work of indian mathematicians who discovered the quadratic formula in the 9th century

[@ForStudy-]

sridharacharya🤫🧏

[@Philip-qq7ql]

Proof?​@@arushjais

[@GaidencioBandi]

Who cares

[@Dr_piFrog]

Because it is Derived as a solution to the Quadratic Equation, could this possibly why it works?

[@YKRotter]

Nice video❤ quality content, u gained a new sub 😀

[@B5Scheuert]

Great video! Just some crutique I hope you'll appreciate: While the animations are great (and imo pretty), they're aittle too fast to follow. I personally wouldn't mind if you paused your talking a little while the animation goes, if that means I can follow each term or expression with my eyes as it moves around. I say this because every time you even just made algebraic manipulations, I had to pause the vid bc the original equation was gone and I didn't quite catch what you did

[@ThomasEdits]

I agree 100%, would like to add that itˋs a little confusing that the terms *don´t* actually move around as the animation just morphs it completely, would be a lot clearer if the terms actually moved around

[@B5Scheuert]

@@ThomasEdits Yeah, I noticed that after writing my comment but didn't bother to edit it

[@virtua_t4695]

Nice video! I liked the part about the symmetry, never thought about that.

[@klausolekristiansen2960]

The reason the general was not used before is that it uses negative numbers. even after it was known, many mathematicians refused to use it because they rejected negative numbers.

[@mahsn818]

2:22 Why is it 2a? and why is c gone?

[@robertthomson1587]

Because it's the solution when you complete the square of the general quadratic formula.

[@woodencentipede260]

At 3:35, why wouldn’t you write it as +- 2a as the denominator for the right hand side after taking the square root?

[@ManojRajput-zf3um]

Ahm *Bhaskara casually spits out qudratic formula

[@chrishelbling3879]

When you say, that the function intersects the graph at any point "where x equals zero," don't you mean, where "y" equals zero?

[@olivermathiasen3594]

@@chrishelbling3879 "The graph" u then mean the horizontal or vertical part.

[@inactive2766]

Your channel is a great work thank you

[@TheChillennial]

I did enjoy this video, and liked it. We were never taught this. Thank you. I have a new found fascination with substitution.

[@ThomasEdits]

Good and interesting video, but it would be easier to follow the algebra if you showed the previous steps as well instead of a morph animation to the solution. the morph it also tricks my brain by giving false flags, such as it moving around when nothing actually changed, or the animation showing terms «moving» one way while the algebra actually did something different

[@Twobarpsi]

I was an A student on high school. Many years later, I can't remember any of this.

[@FundamSrijan]

_"How does the quadratic formula works ?"_ _"It works"_ - **SCHOOL** teachers

[@Akuntest-m2c]

Why when you add those -b²/4a² it doesn't get add to the other side too

[@JahanaraBegum-yf9zv]

that is how maths should be taught … the origin should be understood first and visualised then practicing till mastering it

[@kruksog]

Completing the square! That's where it comes from.

[@gtm5650]

More educational videos please 🙏

[@hudsonbakke8836]

When I was in middle school I derived the quadratic formula on my own because I was lazy and tired of completing the square and wanted something I could just put into desmos and give me the right answer. Then in the next chapter we covered it

[@malabikasaha2452]

Parabolas that don't intersect x axis have complex roots. But parabola cannot be drawn in the complex plane.

[@carultch]

You need a 4-dimensional plot to show a parabola in all its glory, but unfortunately, we're limited to 3 dimensions. There is a solution called Riemann surfaces to take advantage of color to do so. The x-y plane is used as the complex plane as the input to the function, the z-position tells us the magnitude of the output, and the color tells us the angle of the output. Here's the parabola in all its glory. blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLAZCbpJDo8WzwZFUIQPPp-4aoB5LNCqki3Wj2Wz17EdZP_hi57da85IJg0EzsKI76qNN8hsm8FQVjzJkMk7Jxw7zfEDA-fKb4QajrjBOZDk2AOF_ZBu3ZWRu5mcCSWyB3shSHlgawUZd6/s1600/Screen+Shot+2019-12-04+at+10.43.36+AM.png

[@rkm823]

Two equations for finding value of 'x' and 'y' 1) ax + by = C 2) px + qy =C (If "C" is the Real Number as 0,1,2,3......). Can we write or say. "ax+by = px+qy = C" ("ax+by = px+qy") ( Because both are equal to "C"). Is it true or not. please help with example 🙏

[@pi_squared2]

Yes, this move is allowed. For example is x + y = 0, and 2x + 2y = 0, you may write (x+y) = 2(x+y) = 0.

[@JustTruth-x7g]

Amazing ❤️‍🔥❤️‍🔥

[@Aditya-tt2jz]

Well explained 👍

[@olivermathiasen3594]

Honestly amazing

[@memesmemes3143]

SHRIDHARACHARYA QUADRATIC FORMULA 🕉️

[@Vampire-Catgirl]

It would've been nice if you explained completing the square

[@iMíccoli]

I think a nice way to see the complete the square idea is to compare x²+(b/a)x with a perfect square. For example (d+m)²=d²+2dm+m² so: x²+(b/a)x=d²+2dm+m² and it's easy to see that x=d because both are the second degree term . Now we see that (b/a)x=2dm but we know that x=d so b/a=2m therefore m=b/2a ❤.

[@liudvikasvelykis8576]

2:21 from where did that formula come out of?

[@IsakLevana]

Since (a+b)^2 = a^2 + 2ab + b^2 , you can substitute a for x here and b for b/2a to get the second equation. To implement the second equation into the first, just subtract b^2/4a^2 from the square since it’ll retain the original value of x in the equation.

[@derpyyyyyyyyyyyt]

where did you get the 2nd equation from?

[@jmadratz]

Solving the square method was taught in 1st year engineering math courses. Nothing new here folks.

[@powderedphantom5765]

y = 0 not x

[@curtisnotestine3134]

Glad someone else caught that mistake.

[@FlareDevGaming]

It intersects at the line y=0, not x=0.

[@boinks1204]

I don't get it 😢

[@pragyanpranay3681]

2:26 Bro did not use \left( ight) in latex 😢

[@tanmaykulkarni1540]

respect for acharya shreedharacharya for giving the quadratic formula❤🙇‍♂️

[@liar123liam]

What app do you make your videos on?

[@pi_squared2]

This was all animated using Python using 3b1b's library called Manim. The actual editing on this video was made on iMovie.

[@the.real.ihab1]

ok this is way better

[@fcolecumberri]

You can solve with powers of 5 and more, but you need numeric methods.

[@erichnoll874]

I thought the QE works for the same reason most of us do, we can't afford to retire yet.

[@engsherif4657]

Good lecatcher❤

[@b1j]

You’ve made a simple algebraic simplification mystifying and hard to follow. First you gloss over the crux of this solution, completing the square (completely obscuring how it works), and then you blend two forms of a later version of the equation in a way no one can follow. Just try “solving for x”.

[@FatherGapon-gw6yo]

It actually only works for odd numbers.

[@fonebonedon]

Why shouldn't it work?

[@ianfowler9340]

I thought everyone was basically taught this way. I know I was and that was a long long time ago. And I spent 34 years teaching it basically this way. With minor variations of course. So what's happened in the last 15 years? Nice video though.

[@pianomaster30003]

0:35 y=0 not x

[@striker8380]

You mean f(x) =O

[@pianomaster30003]

@@striker8380 same thing

[@striker8380]

@@pianomaster30003 not when the axis is labelled f(x)!

[@Dawaulsy]

4 more seconds and this video could've been roughly 2π long

[@88Conqueror]

Shridhar Acharya formula🙏🏻

[@DinsDale-tx4br]

2.22 lol @ not explaining 'completing the square' .. that's the whole bloody point of it in the first place.

[@DivyanshJha-i9n]

It is shreedharacharya formula

[@ciaverifiedbadge4355]

Pajit formula

[@kevindickin7298]

a non mathmatician would probably say " what the hell is this person talking about".

[@cj-cv7zv]

Not a great way of explaining it for beginners. I know if I didn't already understand the math that this would have only confused me even more :/

[@naomiparsons462]

Yeah, you need to understand completing the square before understanding this.

[@theeswinkler4998]

This is pretty elementary math though, the assumption here is that you've already learned this and are looking to expand your knowledge

[@Thrillzrobloxbedwars]

True

[@Thrillzrobloxbedwars]

​@@theeswinkler4998 It's not

[@SoThisIsMyCat]

Cool

[@bradleyhandlerisfat]

what the hell is this minecraft enchantment table looking formula 😭

[@brianjacob8728]

y=0

[@johnm5928]

Tbh that didn't really explain much. Lots of equations and algebra but nothing tangible 🤷‍♂️. "Completing the square" sounds like a nice visual but alas. What's the curve for? What is it graphing? Why does the quadratic help anyone? Those are the questions that need answering, not showing how/why the equations works entirely within an algebraic context. Math is more than that.

[@lastdeat3286]

Then search that up this is proofing of the quadratic euation

[@williamhorn363]

I disagree on the "why does the quadratic help anyone" point. Math does not have to be immediately applied to be useful or fun. There is an entire degree dedicated to pure mathematics for this very reason. The only critique I'd give on this video, is that I think he should have explained the completing the square step even though he said it was beyond the scope of the video. If you're going to do an algebraic derivation, then you should at least present a baseline walk-through of all the steps involved. Everything was fine to me.

[@johnm5928]

@@williamhorn363 yes but the quadratic IS immensely applicable and useful, and is taught in schools for that very reason. Those who wish to pursue more abstract mathematics can do so, but many people will never succeed in school because basic concepts such as this are not taught well.

[@williamhorn363]

@johnm5928 And I agree with that, but that doesn't bolster the topic of his video, which is how the equation itself is derived. To be honest, the applicability of quadratics is oversaturated as-is, so anyone shouldn't have any difficulty finding that resource online with one search if that's what they wanted to learn. It would have only fluffed this video past its relevancy to the title. This is all my opinion, of course. I respect whatever position you take on this.

[@ایرانی-س7د]

درود بر شما

[@starpeep5769]

FINAKLY A EXPLANATION but i hate being low iq that i can't even come up with this itself let alone solve mechanical/application type of complex anything

[@MikeB3542]

Check out blackpenredpen's (Steve Chow) video on completing the square...not only does he show the algebra, he gives the geometric explanation that makes the process intuitive. kzbin.info/www/bejne/fnvEqnushctknNEsi=GgCoHnA5l210eLms

[@renadnasr7091]

yeah, about history why didn't you mention the inventor of such brilliance?? why not mention Arabs and Al Khawaryzmy?? Today's algorithms are named after him where khawaryzmeya in arabic means algorithm, mind you

[@gamingwithairavat]

Indian always great

[@ViggoP07]

Use the pq formula much better

[@Laplace148]

you cant explain the quadratic formula without explaining completing the square

[@tsunningwah3471]

runge

[@jagarnathprasad5862]

It is shridharacharya 's rule derived by INDIAN mathematician

[@scottydu81]

I still don’t understand 🤷‍♂️ 1. You said to start with ax^2+bx+c=0. Why? What’s the significance of that specific equation? 2. You said “that’s enough history” but it wasn’t. What real-world problems did this equation solve? What happened to whom that made them come up with this? This exact formula is the reason I flunked out of algebra in HS and was sent to remedial math. It just made no sense, it was random gibberish I was just supposed to “accept” with rote memorization. It was infuriating.

[@enderguz3213]

ax^2+bx+c is the general equation for a qudratic. a, b and c are just numbers.

[@scottydu81]

@ What is a quadratic?

[@wowwow5311]

quadratic is basically a (polynomial) equation that contains a second degree​, due to x^2. @@scottydu81 in fairness, to understand this video you need prior knowledge in the first place

[@wowwow5311]

thats the issue with just maths in general when taught in school. you're taught all the formulas but not shown how or why it works. the reason for that is because for the sake of simplicity. generally formulas can get complicated and the proof behind them is also even more complicated. that's why we just memorise them just to pass maths

[@4XLibelle]

I agree. “Explaining” how or why something in math works by merely rearranging it is not explaining it, not at all. I loathe the arrogance or those who teach math.

[@roninkegawa1804]

If you teach math like this to your students you must have extremely confused students. That video is a perfect example on how NOT to teach math.

[@cooIcat6778]

Imagine questioning the quadratic formula. It works because it is the general quadratic equation equal to zero solved by completing the sqaure. If you clicked this video you probably are young and underaged.

[@JatPhenshllem]

Wow thanks. We absolutely did not want to know how the x could be isolated on one side. We absolutely only watch videos on topics we doubt and question.

[@_Heb_]

No one's watching this video because aren't sure _if_ the quadratic formula holds, they're watching because they want to know _why_

[@cooIcat6778]

@@_Heb_ pretty sure you would have seen the proof when learning but okay

[@rustyjona7384]

@@cooIcat6778no, 9th graders aren‘t taught poofs like this

[@emilioozuna4748]

Ok retard

[@jkzero]

very nice, make sure to submit this to #SoMEpi

[@СергейИванов-ы1п8э]

This formula is right cause if you put it into the equation it will be automatically solved