"New" Way To Solve Quadratic Equations That Everyone Is Talking About

This is an incredible problem solving technique. It is amazing the method is not commonly taught.
Original sources - Professor Po-Shen Loh explains the method, history, and gives examples in the following articles and videos. They are worth checking out.
www.poshenloh.com/quadratic
www.poshenloh.com/quadraticde...
www.poshenloh.com/quadraticre...
arxiv.org/abs/1910.06709
• A Different Way to Sol...
• Examples: A Different ...
News stories
www.technologyreview.com/s/61...
www.popularmechanics.com/scie...
Similar methods
The Quadratic Formula, Revisited (Girls' Angle Bulletin Oct/Nov 2017)
girlsangle.org/page/bulletin-a...
Quadratic equations taught in Germany
math.stackexchange.com/questi...
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[@MindYourDecisions]

Here's a video I posted 4 years ago, when the channel was just starting: The Quadratic Formula - Why Do We Complete The Square? kzbin.info/www/bejne/e3PFpaJ8gq-pfMU I was still new to making videos as the channel was just starting. It took so long to do the animations, and I recorded the entire video in a single take so the video is not as polished. So I was overwhelmed by the positive comments, like: "This video should be presented in every high school and middle school algebra class. "

[@akashmukherjee2405]

Great video and animations. Helpful for my cousin. ♥️ It's easier to remember than Brahmagupta's formula, but one thing, in engineering level, this method will take a hell of a time to solve.... Brahmagupta's formula will help at that moment. Brilliant idea anyways.

[@trailokyatripathy4341]

Keep the initially assumed values of B and C in the final result and you will come back to the same old quadratic formula

[@akashmukherjee2405]

@@trailokyatripathy4341 yep.

[@vasilisanagnostopoulos5198]

MindYourDecisions These come from the vietta formulas . They’ve been known for years 🤷🏼‍♂️

[@moeenuddin6756]

MindYourDecisions I wish I knew this 2 year ago

[@manishameruliya2011]

In India (and many other countries) there are three ways to do it: 1. Split the middle term 2. Completeting Square Method 3. Quadratic Formula

[@SantoshSahni23]

They( American ) just changed our well known Quadratic formula 😅 . For just getting some credit !

[@xdgaming952]

i think that's world wide

[@lakshyasingh1576]

Correct

[@yasirarafat7654]

I hate completing square method

[@mrnobody3758]

@@yasirarafat7654 it's ez just watch some yt vids

[@aczajka74]

This is literally a derivation of the quadratic formula

[@bramkivenko9912]

If you divide out the equation so "a" is 1, then the quadratic solution is: -½b ± sqrt(¼b²-c). The innovation is the normalization of a=1, means you can solve quadratics faster. This is presumably important for timed math competitions.

[@nightsky8191]

I thought common derivation of the quadratic formula is by completing the square. Of course the two methods are identical but at least I’m new to this

[@Felixr2]

@@bramkivenko9912 The thing is, in *both cases* you're dividing two numbers by a. The only difference is if you're dividing by a at the start or at the beginning. This isn't any faster.

[@bramkivenko9912]

@@Felixr2 I understand, but I think a person can perform this faster while less likely to introduce errors. To each his own.

[@R_samurai779]

It is the other way around

[@kabochaVA]

The whole comment section: This is "new"? Here in Germany/Russia/India/Greece/... we learn it in 8th grade. It's known as the "pq-Formel" / "Viète's theorem" / "Middle Term Split" / "S(um) and P(roduct) method"...

[@shyamparihar4071]

yup middle term split here in India....we learn it in 8th Standard and then we learn quadratic formula in 10th Standard....as we cannot solve every quadratic equation by middle term split (easily).

[@zynade9334]

@@shyamparihar4071 how is this the same as "middle term split" method? He literally used that in the starting of the vid followed by the "new" method

[@abhi_shek1196]

Also here in Nepal

[@maratimus]

And China

[@justinf8091]

I’m in Canada and we learnt this in grade 7😂😂

[@chilllife5694]

Old: chicken produce eggs America's latest: eggs are produced by chicken .

[@HKClasher]

🤣🤣🤣🤣🤣

[@patrickstar9151]

In Europe and US our education is not that hard

[@fizixx]

You know it's a Chinese guy that came up with this, right?

[@shahnaz18]

😂 🤣 Lmfao... Underrated

[@localtitans4166]

What does that mean?

[@algc19]

This is literally Diophante's method, step by step. "Ancient mathematicians didn't do it like this" well yes they did

[@technicalgamers7324]

No

[@plaustrarius]

Lol tartaglia, Ferrari, cardano, lagrange, they all knew this and did it better honestly hahaha

[@daywill8849]

Shri dharacharya invented this... Not some diophante guy

[@aayushpatra3823]

@@daywill8849 They are talking about the new method, not the old one

[@phoenix0278]

@@daywill8849 hey@Daywill,He do not invented it he discovered it.....AND he is not the only one...😇

[@lowlifexysvendetta5894]

I have known this formula as the p q formula And yes, I am from Germany

[@rishav-singh]

Exactly bro

[@frodobeutlin1564]

Here in Germany we learn it as a quite simple formula in 9th grade. I learned it in 1999 and still teach it that way.

[@geniegerny]

Yeah actually we learned this before the other formula, but we only used it when there was no coefficient in front of x^2. They didn't tell us we could just facture out the coefficient :(

[@NetAndyCz]

Same in Czech republic:) p,q as well. And imho it works well only on nice behaving a, so I prefer the standard formula anyway.

[@kariminalo979]

Sweden too bro, we calle it PQ-formeln just like on Germany.

[@gabrielnobrega1428]

It ends up being the same thing, except that instead of memorizing a formula, you're memorizing a method that ends up in the same formula

[@jamieg2427]

Precisely how I feel. I would rather memorize a formula than a bunch of steps. I feel most students I taught this to wouldn't remember the motivation for most of these steps and instead just memorize. It's better to just teach how to derive the quadratic equation, with variables and with numbers.

[@ravindrawiguna8681]

@@jamieg2427 but with the method we can clearly see the logic of solving it and it more intuitive. If we use formula that we memorize, we only can solve it but never had the deep understanding of why that formula works in the firsr place, and that is just like a robot,like here are the number now crunch that into this formula. I prefer the concept so that when we found a harder or unique problem, we still can solve it and not run out of ideas because we dont have a formula for it

[@Zalamandar]

Well of course it is. All methods must eventually lead to the correct answer, but some methods/formulae might just be easier/quicker/more intuitive to use for different people.

[@shiiqu8174]

@@ravindrawiguna8681 Most students won't need to have a deep understanding and they don't want it either.

[@Etc2496]

@@ravindrawiguna8681 I mean that's why you learn where the quadratic formula comes from how it's derived. But after you see that there's no need to pretty much derive it every time you want to use it. Just apply it directly.

[@sarthakmirani6121]

Presh : Mathematicians found a NEW way for solving quadratic equations The Comments : Nope

[@lunaticluna9071]

"Mathematician finds a new and easier way to calculate quadratic equations" Yeah no it definitely wasnt him that found it

[@Ignasimp]

And it's definitely not easier lol.

[@deivisony]

@@Ignasimp y tho

[@genieinthepot2455]

....It was tho

[@brightblackhole2442]

@@genieinthepot2455 however important they were, they did not invent factoring

[@SK13736]

In the end, we end up with the same process but in a different route.. but it does look more lengthy than the usual way tho..

[@MathswithMuneer]

Very true.

[@zecuse]

My list of steps for both methods, separated into trivial parts: Formula Video 1. b^2 Divide a 2. 2a -B/2 3. Twice the result * c Square the result then - C 4. Square root the difference with step 1 Square root the result (roots are obtained here for both) 5. Simplify the fractions if applicable Notes: Steps 2 and 3 of the formula can be very quick but are necessary for each result. Whenever I've had to deal with the formula in (grade) school, I was always told to simplify the result as much as possible to receive full credit, otherwise why do any of the work to begin with. My college math courses didn't care about simplifying unless it was explicitly stated. Dividing the a can be done across steps 2 and 3 of the video method. When your work needs to be shown, there's fewer places for error because you've already performed the simplification along the way to the answer. You'll notice the steps are the same for both methods, but the division is done first instead of last.

[@NirateGoel]

Expect for the -b you were unfair with steps 2&3 4ac is one step not 2.

[@zecuse]

@@NirateGoel 4ac is the same as doing 2a then that result * 2c. Again, I broke things down into trivial parts that mattered, hence why the subtractions are combined with the roots step. Calculating 2a is a necessary step in calculating 4ac, whose value is also required elsewhere in the formula. I even stated in the notes that this would be a reasonably quick computation compared to all the other steps.

[@aaash2952]

Americans- Whoa that’s a new way to solve Quadratic Equations ! Meanwhile in India- Grade 8 students solve via this method.

[@cullinan1611]

Bhai konse school m ho ?

[@night23412]

grade 10*

[@kavithajagdeesha8999]

@@night23412 grade 8

[@yldrm5704]

@@kavithajagdeesha8999 10 for cbse

[@ishanjain922]

Bruh I'm in 8th and I do the middle term split

[@tomsharpe2185]

This is just 'completing the square'. Completing the square for a general quadratic equation is precisely how you derive the quadratic equation. Soooo... nothing new here.

[@Abhinav-ss3th]

Check again. It is the individual steps that is used in the process. 'Z' is a new variable in itself. completing square uses number functions and the discriminant method to get the roots

[@darthmath1071]

@@Abhinav-ss3th no...unless we have a different definition of "completing the square"

[@stelladavis1798]

Tom Sharpe it's a little different than completing the square

[@_-_-Sipita-_-_]

The original is like finding a product and a sum And the new one is finding a power 2

[@telaferrum]

They both work, but this is definitely not completing the square. 2 ways to get the same answer, but I like this one more since it builds on what we know about factoring. To complete the square you try to get a polynomial from standard form to (x + n)^2 + k Point being that (x + n)^2 is a completed square, so you can move k to the other side and take the square root. With this method instead you want a polynomial to go from standard form to something like a factored polynomial. (x - r1)(x - r2) Multiplying that out to standard form gives Vieta's formula which gives a straightforward 2 equations and 2 variables -B = r1 + r2 C = r1 r2 That's made simpler by showing the roots are in the form r1 = -B/2 + z r2 = -B/2 - z So then it's just 1 variable and 1 equation.

[@maon7565]

A: The quadratic formula is not difficult. B: This method isnt easier.

[@valeryrus]

It is easier if u get good numbers

[@anmolsekhon3545]

its been there for years its easier and in india its more famous

[@VivekKumar-td3me]

@@anmolsekhon3545 which one is easy for quadratic formula is easy

[@KerrankiSuomee]

This one has less to remember and it's easier to use without writing anything down

[@puppetsock]

It is harder. It’s the old method with the added thing of dividing by a.

[@anurag95sharma]

The quadratic formula is easier to memorize. This method(in the video) is intuitive(easier for beginners) but still requires memorization of steps. The beauty of quadratic equation formula is the √(b^2 -4ac) part of it: This part tells you whether the roots are real, imaginary root or single root.(Discriminant) This formula also gives us the shape of the parabola that the equation forms. For beginners memorising the formula is difficult but as you dive deeper into mathematics, this formula looks elegant.

[@extremeextraordinary5588]

Nerd

[@Lolwutdesu9000]

Err, every parabola has the same shape. The only difference is that they are scaled with different constants.

[@ashishkale5424]

b^2-4ac is called as discriminant not determinant!!

[@kariminalo979]

It's called the PQ-formula and it's way easier to remember than the Quadratic formula. You re-arrange the second degree polynomial into the form of: x^2+px+q hence to get the roots, simply plug in the coefficient "p" into the main pq-formula which follows below: x=-p/2±√((p/2)^2-q) If p < 0 then you'll have p/2 before the square root, it doesn't matter what what the whether p is negative or positive in the square root since it will always become a positive number after being squared in side the square root. If q < 0 then it will be written as +q in the square root and if q > 0 then it will be written as -q. For more detailed overview just check out the link of an image that I have attached below: images.app.goo.gl/fEm2U8BJXsa48Z7j9

[@anurag95sharma]

@Adam Romanov My point was it shows where the graph cuts on the x-axis. I thought it was implied in my answer. My bad.

[@scottcarothers837]

Y’all like “only Americans didn’t know this” but i got taught this before the quadratic formula

[@gunjannagda9526]

True

[@urisap9884]

Same lol

[@sjorsbarten8900]

Me2

[@buggyfifa]

Same lol A and B were alpha and beta.

[@siddharthdoshi4858]

Lol same they taught us this method in 8th grade and quadratic one in 9th grade ig

[@FROFilmsIRE]

When you say "everyone" is talking about this new formula, presumably you mean the tiny minority of us who go to KZbin to muse about maths.

[@SeeTv.]

3:45 Nice! You just derived the pq formula!

[@lunaticluna9071]

This is the pq formula.

[@V-for-Vendetta01]

Bruh

[@takvacs]

I'm sorry to be butthurt over here but this is just like that time I thought I had figured out a general equation to calculate the sum of natural numbers only to realize some Gauss dude did that a few centuries back. All the more power to anyone helping people that don't yet know this. Lord knows math needs to be made fun for all.

[@cizma27]

When I was 12 we started doing geometry and learned pythagoras. I realized about connection between angles and opposite sides. I had full notebook of my findings, mostly ratios between longer and shorter side of right triangle. I also had some breakthroughs when I started drawing circles around right angle triangles. My teacher told me that I was on a good path and too keep working. It lasted for a month or so. 2 years later I started the high school, we started doing trig and it all flashed back to me. Basically I (kind of) reinvented tan function as a 12 year old.

[@aarushrathore1276]

As a child what i did Sum of infinite nos is -1/12 and also general formula is n(n+1)/2 and equal them.

[@darkseid856]

@@aarushrathore1276 lim(n -> infinity) {n(n+1)/2} = -1/12

[@Preeesco]

I remember that some years ago I had discovered that a²=(a-1)²+a+a-1 Eg 6²=5²+6+5. Then I realized it was just an "application" of (a+b)²

[@mdzaidhassan8996]

Me too as 7 years old figured out to calculate sum of continuous natural numbers by adding first and last term and for others..

[@bobengelhardt856]

"You don't have to memorize anything" Except a multi-step process with much more opportunity to forget some little, but vital, step. Way, way harder than just memorizing Brahmagupta's formula!!

[@theterribleanimator1793]

At this point you might aswell make an algorithm for finding the root for any type of function.

[@nojomyth]

I find it way easier to remember as - p/2 +- sqrt( (p/2)^2 - q) xD It's also way faster to tell if there's a solution in R for the equation without the need of a calculator since u can just approximate it x^2 + px + q

[@piman9280]

I was always told it was called *The* Quadratic Formula.

[@user-mc8wg6qq3b]

@@piman9280 Hmm, it was discovered by that man, so its sometimes also called brahmagupta's formula

[@eliahabib9267]

The "New Way" in this video is what they teach us in school in 9th grade.

[@ryanmajumder9569]

Ha ha ha sooooo true..👍

[@kepler4192]

Exactly

[@ex_why_zee9126]

It was taught in 7th to us

[@siddharthdoshi4858]

Yeah I remember I used to do it this way in 8th standard. My tuition teacher had taught me this and I clearly remember it was 8th grade coz that was the first time I joined a tuition. Those days were great!

[@alifyasadikot6797]

@@siddharthdoshi4858 agree🙋🏻‍♀️

[@detectivl5811]

This method really only is new for Americans 😂

[@detectivl5811]

@@C4pt41nN3m0 Yeah cause this method is way easier to remember and can be used in more situations 👌

[@frankjohnson123]

It’s not taught because it’s just the same method with more room for error

[@angelmendez-rivera351]

frankjohnson123 That makes absolute no sense. If two methods are identical, then they have the same room for error. That is implied by the definition of identical.

[@Last_Resort991]

Videos like these really imply that americans aren't very intelligent 🤔

[@GD-ep5fl]

@@C4pt41nN3m0 esatto, si fa quando B è pari

[@zwest808]

"an easier way to find the roots." :/

[@mwexim7132]

Zack West I agree. The ‘common’ teached way is easier to memorise. This formula is indeed easier for beginners, but you’ll need to meorise more steps, and that’s not what mathematicians want to do.

[@zwest808]

Mwexim yeah. There are already so many easy methods to find x for quadratics. If they’re going to find an easier way of memorizing types of functions they should do it for higher degree polynomials

[@Ibakecookiess]

@@mwexim7132 This way at least you understand what's going on. Memorizing a formula isn't learning much...

[@lukamitrovic7873]

@@Ibakecookiess you don't really understand much lol. It's literally the way you derive the formula, nothing else

[@mwexim7132]

@@zwest808 The problem is, it is proven that for 5th degree polynomials, you can't find a method for discriminants, so that's indeed interesting.

[@pruthvishthakkar151]

Mathematicians: This is new way of solving quadratic equations Indian teachers (teaching this to students for decades): Hold my chalk

[@ashishmeshram5057]

Indian teachers op 😂😂

[@redskull4447]

Yah -b+_(b²-4ac)^½ -------------------------- 2a

[@bishiyarudesu]

@@roastedpeanuts694 this is a certified good classic

[@krishanuchattopadhyay7006]

@@wockhardt69 And still 45% of your NASA employees are Indian, PM of UK is Indian. 😂😂😎😎

[@krishanuchattopadhyay7006]

@@wockhardt69 When you google remember Sundar Pichai an Indian is the CEO of it. Whenever you count money remember 0 is given by Aryabhatta an Indian. Trigonometry was given by Indians, Baudhayan theorem was taken by Pythagoras and took the patent. So, stop lecturing us, India is the best. AUM 🙏🏻 MAY BUDDHA, NARAYANA, SHIVA give you some buddhi (intellect) to think 😁

[@yashrawat9409]

In India you never learn the quadratic formula till 10th class (high school) Before that the children are taught the product and sum method, but not motivated to generalise it

[@phamnguyenductin]

Here in Vietnam, students learn the quadratic formula in the 2nd semester of 9th grade. It's pretty close.

[@devansh3700]

Yeah and this method is called splitting the middle term

[@tumhregfkahusband8725]

I am from india and i learnt quadratic formula in 9th grade

[@humaneBicycle]

@@tumhregfkahusband8725 he is talking about cbse and not icse or state board

[@dinul1719]

Same here in Sri Lanka

[@skwbusaidi]

Inteseting mthod . But if the forget the quadratic formula, I will do completing the square which is not harder than this method x^2-8x+15=0 x^2-8x=-15 x^2-8x+16=-15+16 (x-4)^2=1 x-4 = 1 or x-4 = -1 x=5 or x=3

[@avrilcadeul3207]

This is the method that should be in the video ! 😂

[@ugurcansayan]

"Complete squares" is like the ancient proof for life, time and everything! :)

[@karinano1stan]

yeah, our algebra professor showed us that way

[@Raghad-hc7nn]

Sami can someone explain where the 16 came from?

[@Raghad-hc7nn]

Sami is it a method to make -15 become 1 or 8x2

[@timoriusmaximus]

Here in Germany this method is very well known as the "pq-Formel" ("pq-Formula") and definitely nothing new.

[@theraytech54]

It's only similar and he talks about it

[@koldovalnya]

Or "Vieta's theorem"

[@kallewirsch2263]

@@theraytech54 It is not similar. It is exactly the same. Just all the individual steps combined into 1 formula. The derivation of the pq formula however is usually thought in a more geometrical way. But if you think of it, what is done here is not that different from "completing the square" in an algebraic way. I am not to say, that this derivation is bad or anything like that. It is a nice derviation. But to call it "new" is way beyond what it should be called.

[@randomguy8461]

This is just quadratic formula with extra variables and steps, @4:09 it's literally exactly the same thing only it has been chosen to write C in place of c/a and B instead of b/a. I feel like I would be less upset if this wasn't called "a new method" (i.e a way to completely avoid the quadratic formula) but instead called something like "a new way to derive the quadratic formula" or something similar.

[@TheScienceGuy10]

3:56 Substitute the value of B and C as -b/a and c/a and voila, you have the quadratic formula :)

[@shade5554]

Bruh, I was so utterly disappointed watching this video, thought he was gonna talk about some revolutionary way of finding roots

[@johndoeanon445]

When I saw this, I thought for a moment: "I wonder if it's just going to be a complicated way to do the PQ-formula". A few minutes later, lo and behold.

[@nowonmetube]

To me actually the beginning formula looks exactly like the pq-Formel, say what? O_o

[@RegorForgot]

God damn you and your pun

[@OriginalSuschi]

nowonmetube ah I love the PQ-FORMEL (speak it out loud in a very german accent). Ich mag die Mitternachtsformel mehr als die PQ-Formel (auch wenn's irgendwie das gleiche ist...)

[@AShu-ey9zu]

Thanks sir. Your videos are great inspiration for me and my students to explore new horizons of maths as I have been also involved for preparing students for class8 maths Olympiad at my school level and we have won last year gold for our campus amongst 18 campuses participants.

[@e1coachingcenter]

Nothing new, already using it for last 3 years

[@Tusharsharma-lm8bd]

So you took your content from such places and then teach that to your students on your name It's a shameful act

[@goutammandal2968]

Hii sir

[@Kanoemirate]

This approach looks more time demanding and it's not a new thing. I learnt in back in Secondary school in late 90s. And in Nigeria🇳🇬

[@tamirerez2547]

I gave a LIKE just because the last line: and in Nigeria...

[@StRanGerManY]

@@tamirerez2547 mentioning Nigeria makes any comment funny!

[@kobe11111]

I have a question. Where exactly is this regarded as a new method? Because I learnt this exact technique 5 years in high school...

[@MrHatoi]

Where did you go to high school? I'm pretty sure they don't teach it in most parts of the US but they do teach it in some countries.

[@kobe11111]

@@MrHatoi I actually went to high school in Jamaica and that's how they teach it across the island.

[@MrHatoi]

@@kobe11111 That's interesting to know. I went to school in the US and I never learned this.

[@kobe11111]

@@MrHatoi live and you learn I guess 🤷🏾‍♂️

[@eRayPeitz]

I'am from Germany and we get taught the first part of this method under the name "PQ"-Formla. x = - (p/2) +- sqrt( (p/2)^2 - q ) I always were mad that we not got taught about the abc-formula because for some equations the division with the a factor wasn't that easy. This method really helps me, because I like the quadratic approach is rasiert than the root approach.

[@chrisbhaus5813]

I am from Germany. I'm 36 years of age and even my mom learned this "new" method at school. In Germany every student older than 16 knows this method as the "PQ Formula". And this is since...I don't know...Kaiser Wilhelm, I think 😂😘

[@Negs42]

Haha same..I'm from Germany too..

[@madhuawasthi993]

First of all it's not bhram Gupta's quadratic formula it's Shridharacharya's quadratic formula

[@Sailed_away]

Both were from India , so kindly shut up ....

[@pragul1999]

@@Sailed_away well its like saying George Washington and Donald Trump are both america's presidents hence they are the same guys

[@bidishadey3815]

Exactly, this is sridhar acharya

[@akhil3271]

@@pragul1999 ,😂😂😂😂😂 You made my day

[@piman9280]

Better still, *THE* quadratic formula.

[@carrotfacts]

You’re memorizing a whole method that explains exactly what the quadratic equation is DOING. In fact it sounds even harder to memorize as it’s wordier. It’s much easier to know the quadratic equation

[@UltimateBargains]

3:46 I recognized the classic quadratic formula when he got to ±√ thing.

[@ruthsquire958]

No, it's not. It's easier to know a formula but it's better to learn a method. Always. That's because embedded within the method is the formula.

[@SlingerDomb]

Should be great to teach kids about this instead since mathematics is all about following logic and thinking not purely memorizing.

[@AniketKumar-cs9wb]

I already have seen this 5 min ago

[@MindYourDecisions]

In that video at 1:10, Achrotone noticed the typo "back-to-bank" which is now corrected: kzbin.info/www/bejne/onXFgnRqmtp0mLs&lc=UgwYAupZwmfkh8Nfred4AaABAg

[@fix5072]

@Shailesh Kumar true, and the formal description is easy for everyone to do it theirselves. I'm from Germany and we learn this formula from 5th grade (10yo) and I never knew this wasn't taught in the entire world 🤷‍♂️

[@AshutoshIIT]

@@fix5072 yes...& That formula is invented in India....& now used in all over the world 😃

[@adi-sngh]

@@AshutoshIIT ok boomer

[@annapurnarangrej4221]

@Shailesh Kumar hey Bro, already our quadratic formula was derived by dividing 'a'😄

[@Jnoel608]

As an American, I feel like we learn the most complicated unintuitive ways to do everything. Just to be different. (Edit) never mind. It’s really the same as the quadratic formula, just slightly different way of thinking about it. Depends on what you prefer. Remembering the formula vs remembering the concepts.

[@aymanenouhail5241]

the quadratic formula is easier xD it's plug and play, you can't go wrong with it. This one has a higher degree of error and requires more steps. I'll pass. you also get shown the proof for the quadratic formula when you get taught for it (at least here we do) so if you don't intuitively understand it it's your fault for not paying attention in class. Not that you really need to know the proof since the formula is elegant and short.

[@anthonypazo1872]

I had never been taught this before, but now that I have seen it, it seems so obvious. Well done explanation of this process.

[@aniketroy805]

Here is nothing new.....you have just solve it by using shreedharacharya's formula.....just a another proof of the shreedharacharya's formula

[@gaurangrajai401]

Could you plz share any information about it( Shreedharacharya's formula), I'd be grateful to you.

[@Salien1999]

Any method that solves a quadratic equation can be used to prove Sridhacharya's formula, since the resulting solutions will be equivalent. The way we should judge methods like these is by how easy they are to remember and execute. This method, while more intuitive and easier to remember than Sridhacharya's formula, takes a few more steps to do. Therefore, neither is inherently better. They are both useful, for different circumstances and mathematicians.

[@aspiringscientist1616]

Are u from Agartala?

[@michaelz2270]

People should stop calling this a new method. Anyone who has graphed a parabola can see that the roots are going to be of the form m +- n where m is the x-coordinate of the vertex of the parabola, given by m = - B/2 in your notation. Yes, in the centuries people have been solving quadratic equations, people have tried plugging in - B/2 +- n and solving for n. If they know a little more they'd set the product (-B/2 + n)(-B/2 -n) = C and solve for n. Seriously, this needs to stop being treated as something new. All these fancy degrees being shown off just make the situation more infuriating.

[@mousumibhamjasanyal5506]

It's the commonest method of solving a quadratic equation in India.

[@nesha_owo]

when it's Christmas break, but Math is still haunting me from my recommendations :3

[@rankanidhibehera5527]

This is brahmagupta formula this is sridharacharya formula

[@ViratKohli-jj3wj]

@Victor Yago you too brutus

[@paweplewa3644]

When you had known this method before you watched the video *A Genius* But then you looked at the comments.

[@humanityisnumberone6008]

Nope,this is just a simple mathematics at 9th grade

[@paweplewa3644]

@@humanityisnumberone6008 Not in my country ;), It's just a joke man.

[@Icy-ll5ie]

@@humanityisnumberone6008 my teachers have not thought me this method

[@pallabgoswami2451]

Absolutely it I knew this when I was 13

[@paweplewa3644]

@@pallabgoswami2451 Same as I. But in Poland we aren't taught this method, It is not in our schools.

[@thetntsheep4075]

I find it easier to complete the square. Maybe just because I'm used to it 😁

[@williamjohansson1963]

From what I have learned, it's making it unnecessisarly hard, just think of it as x^2 + px + q = 0 , x^2 + px + (p/2)^2 = (p/2)^2 - q (x+p/2)^2 = (p/2)^2 - q x = - p/2 +/- sqrt ((p/2)^2 - q) Its litterally the same method but 10 times easier than the one shown. European countries tend to use this one.

[@feelthereal5742]

Can you make a video on solving cubic equation without guessing any solution. I want you to do that because your explanation is good.

[@vasileiospapazoglou2362]

You don't guess in cubic there are formulas and other methods to solve them.

[@detectivl5811]

x^2+bx+c = 0 x^2+bx+(b/2)^2-(b/2)^2+c = 0 (x+b/2)^2+c-(b/2)^2 =0 That's even easier than remembering the formula, cause this is just common sense

[@siveroo7493]

@@vasileiospapazoglou2362 yeah but that's so complicated, and guessing the roots seems like a better method to go, although not really effective

[@user-xb9yv2ci4c]

There is the cardanic formula for this.

[@feelthereal5742]

@@user-xb9yv2ci4c suppose I forgot the formula. Now there should be a straight forward method to find the solution.

[@realhawaii5o]

I feel like the Quadratic formula is much more useful. Having studied maths until the end of high school, and now having calculus and algebra in university, I feel like having the discriminant is quite useful. As you delve deeper into maths, the formula is more useful than this alternative method which might seem more intuitive for beginners.

[@calebking1608]

You don’t have to throw the discriminant out the window though. This method is simpler.

[@anitagofradump5195]

I wish i knew this method for some of my pure maths classes in diff eqs when the type of roots didnt really matter than much and we were forced to solve them no matter what. (And we werent allowed calculators so i think this method woulda been easier to do by hand than the quadratic eqn)

[@Michael-zn2jc]

R2D2 from Star Trek I prefer the quadratic formula. It’s a formula and it’s good to solve any quadratic equation. I’ve used it so many times it’s almost impossible for me to forget.

[@molamola8305]

This is not any "new" way.It just resulted from the property of quadratic coefficients.Everyone who knows Brahmgupta's formula knows this.

[@KumaKaori]

Cripes, any article I read on this mess was incomprehensible. This video makes it make /a lot/ more sense. Thanks Presh!. I genuinely don't know how much of this process and or the cheat steps I may have been taught in school. Been too long and I figured out how to get my ti-89 to solve the tricker ones for me quite early.

[@whyareureadingthis6719]

To be honest I can't see how this makes it easier, u just took the quadratic formula and simplified it

[@user-xb9yv2ci4c]

In Germany, we learned in school first something we called "quadratische Ergänzung", which could be translated to quadratic completion. So, one adds and subtracts a number, so that the formula becomes (x-d)^2+e=0. In the next lesson, we derived the quadratic formula out of this.

[@user-xb9yv2ci4c]

Fun Fact: We call the quadratic formula "midnight formula" because it is so important, that you must be able to recall it, even when you are woken up at midnight.

[@whyareureadingthis6719]

@wise ol' man yeah that's what I meant to say, but I can't see how it's easier than just knowing the quadratic formula, it feels like u added some extra steps to it

[@whyareureadingthis6719]

@@user-xb9yv2ci4c we never learned something like that here, we just learned the quadratic formula straight away

[@gilbertlondre2497]

0000000 0000000 in the USA (and maybe other english speaking countries we call it “completing the square”

[@TheXzasxz]

I still find the method I've been tought over 30 years ago the easiest to remember and provide. What you need is to always remember two equations: (x+a)^2= x^2+2ax+a^2 and (x+a)*(x-a)=x^2-a^2. Simple example is x^2+6x+8=0 x^2+6x+9-1=0 (x+3)^2-1=0 (x+3-1)(x+3+1)=0 (x+2)(x+4)=0 So we have the solution.You may notice, that if you try to solve a general equation ax^2+bx+c=0 using this method, you will get the quadtratic formula, which was always hard to remember to me...

[@baluandhavarapu5786]

This is basically the completing the square method with a little more clear teaching. At the end, you do reach the quadratic formula. This is just a derivation for the formula itself.

[@CharaNoTame]

This is also known as the PQ-Formula, we were told to memorize it (in my case). However I've also seen students remember this formula with some rules of thumb. (0. make sure the equation is x²+px+q=0, and not let's say 2x² or so) 1. divide the middle term (p) by -2 2. add the +- symbol and draw the square root, then square the term from step 1 and put it inside the square root 3. subtract q (also inside the square root) formula will look something like this x = (-p/2) +- sqrt((p/2)²-q)

[@dhruvbhargav1547]

.

[@VishalMaurya]

I know it, cz im in 10th

[@saikalyan5559]

Even 7th class student know this

[@aashayagrawal1334]

I am doing engineering and i didn't know it 🤣🤣

[@aashayagrawal1334]

@@dhruvbhargav1547 shame on me 🤣

[@aashayagrawal1334]

@@dhruvbhargav1547 This formula is pretty good but doesn't gives us the discriminant which is important. You arts with maths 😧 you are pretty brave

[@policarpo4816]

In Italy we are taught a similar thing as a shorter formula for quadratic equations when b is even. If a=1 (which is in this case done by dividing everything by a) you get exactly that formula. Also, the new formula is not so easier to remember than the normal one

[@zemp9671]

I’m italian too, and i can tell that this is off topic. I mean, it is true that you can simplify the quadratic equation but this has nothing to do with the method explained in the video

[@policarpo4816]

YOLO Zemp the topic is indeed different and we use different demonstrations, but the formula is still the same

[@lorenzot.7045]

In poche parole hanno scoperto l'acqua calda

[@policarpo4816]

lorenzo timpone esattamente 😂

[@gabrielenicosia7121]

Si, la formula del “B Mezzi” con il B pari

[@aaronwarwick9966]

I came up with this method when I was about 12. But not for doing quadratics, I had no idea what they were. I was using it for a "magic" trick that I made up, and this was a quick way to get people's answer. When we learned the quadratic formula I told my math teacher I'd come up with something similar, and I thought it was a tiny bit easier. He said "nah. Shut up and write down the equation". Love of math ended that day...

[@taskmaster4624]

oh boy i feel bad for ya

[@tonksonk951]

I saw this article a few weeks ago, nice to see a video on this :) Though, my biggest qualm with this method is the fact that if the coefficient A is not equal to 1 you have to factor the function into that form which can result in B and C becoming cumbersome fractions. Other than that it's a really nice way to be able to think about the roots and the mathematical intuition behind them.

[@probropalzlive6961]

Or, as Indians are taught, for the quadratic equation ax²+bx+c, find p,q such that p+q=b and pq=ac. Thus, -p and -q are the roots of the equation.

[@aguyontheinternet8436]

fractions aren't that hard to deal with. Math can throw a lot worse at you than a couple ratios

[@scottierna5831]

It is not just similar to what's taught in Germany. It is exactly what I was taught in my German high school. But our teachers explain it easier😂 You can use the stuff in the square root for finding out if the function is a passant, tangent, or secant line as well. In the complex method the factor a is just integrated, but that confuses students and/or lead to small mistakes you do even if you know how it's done.

[@OlavurNonE]

“But the key insight in this method is that you dont have to memorize anything” Yeah sure...

[@justinsir]

You people are using calculator So, No need to memorize tables No need to memorize square root. No need to memorize algebraic identity (a-b) (a+b)

[@leodeiters7486]

4:06 I'm German and here the formula isn't ax*2+bx+c but ax*2+px+q. Then we can build the next formula: X1,2= -p:2 +- the root of ((p:2)*2 -q). If you solve this you get the two solutions. Then you can check them with the "Satz des Vieta" like you showed in the video. X1+X2=-p and X1*X2=q. It's very simple if youve done it often enough.

[@entitatohneinteresse4335]

Actually students in Germany don't just learn the "new method" which we call the p-q-formula because usually we use p and q instead of B and C, but also learn 1.quadratic addition (I don't know the english termus technicus, but it's the method used in the proof of both the "old" and "new" quadratic formula), 2. p-q-Formula 3.formulas of Vieta, which are basicly the expressions for B and C as sum and product of x1 and x2 presented in the video) and afterwards, just for fun your "old" method, but we call it the midnight formula or ABC-Formula.

[@erikkozhevnikov3444]

actually, when you're dealing with parameters and very small numbers which you mostly will it just becomes more complicated, thats why you teach the formula, this method is only good for convenient numbers in easy problems.

[@ajfalo-fi3721]

Exactly, the examples given here were pretty convenient numbers

[@michka841]

Mathematicians : how to solve equations Engeneers : *DØ ÅPRŌXĮMĀTÎØNS*

[@amingholizad]

this is exactly the quadratic equation and how it came to be. when first taught the teacher shows how we came about this formula so we can use it confidently knowing why it works

[@manojmohan9893]

And here I was worried that I missed some easy method of solving quadratic equations in school. I studied in a CBSE school(that's the central education board in india) which is average in terms of difficulty across our country. Some of the states do have tougher syllabus. And I still remember learning this method, which in india we call substitution method for deriving solutions to a quadratic equation( I may be wrong in naming). What I do understand from this video is that there does exist a difference to approach to math education in india to US. In india, we may have relaxed approach towards every other subject. But when it comes to maths we are actually very rigorous towards learning by practice rather than remembering a simple formula. Infact we are usually taught the long method of calculating something before we even learn to use a formula.

[@PartyMain]

This method also allows you to find the middle (aka top value) of the function. I knew there was possibility to find x-values through addition/subtraction of the x value of the top

[@edwinchang2760]

I've seen this method a lot

[@MathswithMuneer]

Thats cool, and how about videos that I have for several topics ?

[@Anathorn7]

You do a very nice job teaching in such an interesting and eye-pleasing way! :) Could you please tell me what program do use for this kind of animations?

[@coleedward9642]

I mean it makes sense. In the Quadratic formula, you dividing by a, same in this situation. If you rearrange the denominator, you get -b/2a or -B/2. Then you're left with +/- sqrt(b^2 - 4ac)/2a. Square/sqrt the denominator and you can combine them, leaving sqrt(b^2/4a^2 - c) or sqrt(B^2/4 - c) or.. z^2 = B^2/4 - c or B^2/4 - z^2 = c. So this method is literally the quadratic formula, it's just reordered and rearranged. Sometimes understanding how to solve something requires a different look at what is being done. Some people understand 2x2 better than 2^2, that doesn't mean they can't solve 2^2, they just understand it better a different way.

[@thecaretaker0007]

I derived the quadratic formula myself when I wasn't able to memorize the formula ... And then I didn't needed the formula to get roots.

[@priyankandatta5644]

Like in icse, we know this in class 7. Thats like when we were 12 year olds.

[@keyboardcorrector2340]

I was reading Wittgenstein and solving advanced integral calculus problems at age four my guy.

[@megacahh870]

@@keyboardcorrector2340 lmao

[@nabeelkhan4910]

Ikr

[@stigastondogg730]

That was my thought - it’s the method of “completing the square” which then works out to be the quadratic formula anyway. In exams when asked to derive the quadratic equation, this is the method I used

[@helobhay9427]

@@keyboardcorrector2340 This is nothing.. We start perp for IIT JEE from junior kg and till 4th we're on top of the world.... Sarvashaktishaali Gaitonde🤣

[@harshraj447]

In India, even 8th class students know about this method as factorization method. We have even made various tricks to make this way ever faster. Hence, nothing new for me. 😂

[@peterprastakos9152]

lol right, let me know when India starts beating the US in the math olympiad

[@random_shit_online6104]

@@peterprastakos9152 🤣🤣

[@Singh-be2qn]

@@peterprastakos9152 good morning

[@aamitanandd]

@@peterprastakos9152 lol let me know when American firms stop hiring Indian CEO.

[@MyReligionIs2DoGood]

@@aamitanandd Nice burn! :)

[@ospreytalon8318]

Here's a much more interesting approach in my opinion. A neat result is that for a quadratic Q with turning point T we have: Q(T+-t)=Q(T)+at^2. If the solutions are T+t and T-t, we obtain at^2=-Q(T) which gives t=sqrt(-Q(T)/a). Thus the solutions are x=T+-sqrt(-Q(T) /a)

[@Kairikey]

But wait......... Looking the -B/2 +/-z, wouldn't it actually also be that z= sqrt(B^2 -4C)/2? Considering that the a of the quadratic formula in this situation is always =1 then this is still the quadratic equation.(I mean, duh) If B^2/4 - z^2 = C, then z^2 = B^2/4 -C or (B^2-4C)/4. Then z = +/-sqrt(B^2-4C)/2. That's an interesting way to think about it. So in the end, this method is simply a quadratic formula through a new perspective that are more intuitive, or simply also a way to make sense of what the quadratic formula actually represents. In other words, quadratic formula can be thought of as the average of B +/- an identity that is related to B and can give a product of C.

[@detectivl5811]

Yeah this is pretty similar to the quadratic formula 😅

[@aryanmadhwani]

I'm solving from this method since childhood 😂

[@ypn.official]

We Indians are a step ahead than the whole world but we a too ignorant of ourselves... *Edit:* I was just trying to be a tad bit patriotic & was pointing out how we ignore our own potential... If the comment appears kinda racist 😑

[@keyboardcorrector2340]

*Galaxy braining intensifies*

[@VivekSingh-ts1ec]

@@ypn.official don't say like that. It seems to be a racist comment.

[@Toryo.]

In italy too...

[@detectivl5811]

@@ypn.official Its not only Indian... It's everywhere but the USA 😂 (In Germany we know this method too in 7th grade or so)

[@AndriiBilous]

Dear author! Where were you with your video 30 years ago when I was at school! That's great method, damn!

[@atharvachaudhari6325]

In this lockdown period this video was one of the best that i have seen for revision 😃😄video was awesome 😃😄

[@dinnerxet]

Me: does this method Teacher:Well yes but actually no.

[@anuragsuresh5867]

In the UK everyone that does Further Maths learn this in a topic called roots of polynomials.

[@MathswithMuneer]

Very true, Hi, I am making Videos for O\A levels as well. Please feel free to check them and do share your feedback.

[@alastairbateman6365]

May I complement Po-Shen-Loh on a brilliant exposition of a bit of basic, fundamental mathematics. A good example of plain honesty, simple truth and easy understanding. I first watched the video a day or so ago and it was only a day later that the penny dropped. As per Leonhard Euler's 'Elements of Algebra' (x-a)(x-b) = x^2 - (a-b)x + ab where as we know a & b are the roots of the quadratic. Taking (a+b)^2 and (a-b)^2 [ i.e. props. 4 & 7 from book 2 of Euclid's 'Elements' ] then expanding and subtracting we get the answer 4ab hence we have (a+b)^2 - (a-b)^2 = 4ab. This is a theorem, prop.8 of book 2 of the 'Elements' and for some unknown reason demoted to a RULE alias 'The Quarter Squares Rule'. After a bit of simplification we end with [(a+b)/2]^2 - [(a-b)/2]^2 = ab. The algorithm given in the video then amounts to [(a+b)/2]^2 - ab = [(a-b)/2]^2 which taking the square root leaves (a-b)/2. So (a/2+b/2+a/2-b/2)=a & (a/2+b/2 -a/2 -[-b/2])=b. It is still highly commendable that the 'QSR' has been derived by another route and has been admirably utilised for the factoring of quadratic equations. What I find a bit astounding and some what sad is that together with the hits on the 3blue1brown and MindYourDecisions videos on the same topic a combined total of around 1,453,000 views no one else seems to have spotted the connection. Finally if we change a & b to x^m & x^n then the answer (ab) becomes x^(m+n) hence all integers raised to a power above the second are the difference of two squares . Further more the bigger the power the more DoS solutions there are for any one integer raised to that power! What does this mean for Fermat's Last Theorem.

[@VictorDiaz97]

If memorizing a formula is the hardest part of the quadratic formula, maybe math isn't for you.

[@MsMumuku]

From my experience as a math teacher, even though students know both methods, most will prefer Brahmaguptas Formula because they can just plug everything in and the solution falls out. They will even use it if the solution basically jumps at them. Mostly students for which math classes aren't a constant state of hardship will take a second to consider what might be a quicker or easier way. Cheers to all the people in the comments who are vocal about finding this trivial. Thanks for letting everyone know.

[@Ni999]

Well said.

[@deadfr0g]

Video length: Eight minutes, eleven seconds My reaction: QUICK MAFFS

[@billj5645]

At 3:27 in the video the righthand side lost the negative signs on B. I realize that this doesn't matter, essentially you left out a step where you multiplied both equations by negative 1, but some people might miss this. Note that the "new way" works with equations where a=1. If you start with this assumption then the standard formula for solving quadratic equations becomes somewhat simpler. If you look at the steps in the "new way" and start writing them down in a way to solve for x then the result looks suspiciously like the standard formula. I did this when I first saw one of these videos pop up on the internet recently. It appears that you took the standard formula, dropped out the a terms since a=1, and moved the 2 from the denominator up into the radical. So rather than consider this a "new way" to solve a quadratic equation, it should be viewed as the derivation of the standard formula. At 3:6 in the video the equations on the righthand side of the screen are exactly the standard formula with the 2 in the denominator moved around. Granted if a person was alone on a desert island and couldn't remember the standard formula, they could use the "new way" to reason through to a solution. I'm an engineer and I have to solve quadratic equations frequently. It is second nature for me to plug the terms into the standard formula to get the solution.

[@prashantdahiya711]

You have changed my life by telling is method.

[@haris525]

Yeah I just convert my quadratic equation into a matrix and solve it 😂 - Gram matrices are a good start if you are interested.

[@adorablewigglingbunnynoses2792]

How do you solve quadratic equations by turning it into a matrix?

[@haris525]

adorable wiggling bunny nose sugar high you can do it - trust me. It’s o e of those magic things linear algebra allows you to do. Also you can do it via change of basis.

[@adorablewigglingbunnynoses2792]

@@haris525 I'm a visual learner so could you provide both an example of Linear Algebra and Change of Basis examples of solving quadratic equations please?

[@haris525]

adorable wiggling bunny nose sugar high please 2x2 check Gram matrix where q(x) = x^t * K * x where x =, or Chlosky factorization . It’s hard to work math out on KZbin because it’s hard to type equations however If you are interested you can find examples online.

[@hhhhhh0175]

@@haris525 why on earth would you use matrixes to solve quadratic equations?

[@abhinavmanawat4916]

At 9th grade we were taught three methods to solve quadratic equations. Out of three ,two are described in the video. If you know the third one then like the comment😁😁😁😁😁😎😎 Hint: alpha squared /b1c2-c1b2

[@HimanshuSharma-mv4sg]

Factorization method (Splitting middle term), Completing square method, and Quadratic formula

[@wolo0048]

I believe this would make more intuitive sense starting with the symmetric property of a parabola, specifically that 2 solutions equidistant from some value m would show f(m+d) = f(m-d) = 0. In other words, f(x) = a(x-(m+d))(x-(m-d)). Then expanding would result in f(x) = a(x^2 - 2mx + m^2 - d^2). So as long B = neg (2m) or m = neg(B/2) and C = m^2 - d^2 or d = sqrt(m^2 - C), then we have our solutions of neg(B/2) (+/-) sqrt(m^2-C). The scaling affect of 'a' has no impact here, as sliding values of 'a' does not affect zeroes. Of course, the assumption of solutions here does require FTA, but Gauss took care of that for us :)

[@IanJefferies]

This is just a derivation of Brahmagupta's quadratic formula. At the 4:10 mark you can substitute back for B=b/a and C=c/a and recover the standard formula with some straightforward factorisation. The one thing I have learnt from the video is how the Brahmagupta's formula is derived. Something that should be taught alongside memorizing a formula (one that is still stuck in my head after more than 35 years).

[@viewer4229]

Sridhracharaya formula 🙏🙏🙏🙏🙏

[@antipro4483]

I just tried to say that name and now the world is upside down

[@viewer4229]

@@antipro4483 it is simple, but not easy!!!

[@_MrMoney]

But... why would I do all of that process when is easier and faster to just do the formula? It's not really that difficult to learn and it's way quicker to remember 1 simple equation that all of that method.

[@zecuse]

Because when I was in school, I had to show my work. "That process" is the same steps for both methods, but the division is done first instead of last.

[@raindrophoh]

I would say it would be important to know how something work. Take daily life as example, yes we know we turn on the stove fire come out, but it would be better to know more about why is there fire or things related like combustion. Another example would be we know stepping on the paddle the car would move forward, but would be better to know about how an engine work.

[@Jessica-ib8ri]

Mr. Money it depends sometimes you don’t always have to do the long formula method just use your head

[@duckymomo7935]

Quadratic formula is computationally faster than ph shen lol this for computers

[@vaibhavm8043]

Absolutely amazing. It's really very easy and will be helpfully throughout. Thanks alot.?

[@sehabel]

I'm from Germany and I never learned Brahmagupta's formula, which is called abc formula in Germany, in school. We learned the p/q formula, which works like this: 3x²+9x+6 = 0 |/3 x²+3x+2 = 0 x²+px+q = 0 p = 3 q = 2 x1 = -(p/2)+√(((p/2)^2)-q) x2 = -(p/2)-√(((p/2)^2)-q) x1 = -1 x2 = -2

[@mango5030]

Literally what we study at class 10

[@paoloposo]

I've been to school in Germany and I can't recall ever having learned a method to solve a quadratic equation that involves guessing. I could never remember the ABC formula when I was in school so I used the completion method and it is still my preferred method today.

[@farazshaikh4967]

You got yourself a new subscriber.

[@boguslawszostak1784]

This is the EXACTLY quadratic formula. The first part is p = -b / 2 * a The second part is sqr ((b * b / (4a * a) -4 * a * c / (4 * a * a)) = sqr ((b / (2a)) ^ 2- c / a) = sqr ( p-2-c / a) In my opinion, the easiest way to remember a solution to a quadratic equation is: 1. Substitute x = u + v 2. Find v for which in the equation on u the coefficient at u is equal to ZERO. example: x ^ 2-8x + 15 = (u + v) ^ 2-8 (u + v) +15 = u ^ 2 + 2uv + v ^ 2-8u-8v + 15 = u ^ 2 + (2v-8 ) + u (15-8v + v ^ 2) if 2v-8 = 0 then u ^ 2 + (15-8v + v ^ 2) = 0 so v = 4 u ^ 2-1 = 0 u = -1 or -1 u + v = 3 or 5 I know, counting is usually more, but the method to remember is VERY SIMPLE and that's why there is a formula to remember, for solving the quadratic equation by count less. anyway, the simplest method "for today" is to enter: www.wolframalpha.com/ and in the window that appears x ^ 2-8x + 15 = 0 and we have www.wolframalpha.com/input/?i=x%5E2-8x%2B15%3D0