Germany | A Nice Math Olympiad Algebra Problem

[@DanielDimov358]

x^2 + (3x/x-3)^2 = 16 x^2 = 16 - (3x/x-3)^2 x^2 = 4^2 - (3x/x-3)^2 (4^2 - (3x/x-3)^2 - difference of squares) x^2 = (4 - (3x/x-3))(4 + (3x/x-3)) After simplifying the expressions in the parentheses... x^2 = (x-12/x - 3)(7x - 12/x - 3) Using polynomial division... x^2 = (1 - (9/x-3))(7 + (9/x-3)) Then let 9/x - 3 = a From this we can express the x = (9 + 3a)/a And we can substitute in the equation to get... ((9 + 3a)/a)^2 = (1 - a)(7 + a) (9 + 3a)^2/a^2 = -a^2 - 6a + 7 (9 + 3a)^2 = a^2(-a^2 - 6a + 7) Expanding the parentheses we get... 9a^2 + 54a + 81 = -a^4 - 6a^3 + 7a^2 a^4 + 6a^3 + 2a^2 + 54a + 81 = 0 a^4 + 8a^3 - 2a^3 + 9a^2 + 9a^2 - 16a^2 + 72a - 18a + 81 = 0 a^4 + 8a^3 - 9a^2 - 2a^3 - 16a^2 - 18a + 9a^2 + 72a + 81 = 0 a^2(a^2 + 8a + 9) - 2a(a^2 + 8a + 9) + 9(a^2 + 8a + 9) = 0 (a^2 + 8a + 9)(a^2 - 2a + 9) = 0 Roots of first equation are: a1 = -4 + √7 a2 = -4 - √7 Roots of second equation are: a3 = 1 + 2√2i a4 = 1 - 2√2i Using the roots of the first equation... x = 9 + 3(-1 + √7)/-4 + √7 = -1 - √7 x = 9 + 3(-1 - √7)/-4 - √7 = -1 + √7 ...after multiplying by the conjugate. Very long and unnecessary I know, however it is the way I solved it and I wanted to share it.

[@sofyanaprisandi8634]

Di ..

[@sofyanaprisandi8634]

Di ..

[@duncanenyodi6222]

I did this kind of maths about 40 years ago. Thanks for the reminder son.

[@mikihayelom-zf9hx]

May orpy

[@michellewelchman9245]

I enjoyed your work and your explanation. Excellent and well done! These problems can be long and it took nearly three pages for this one! Lol Sometimes it feels like you're going down a math rabbit hole with no end but in the end when you do reach a solution (s) it's all good.

[@RejepOrazmetow]

Как прекрасен язык математики. Всё понятно, подробно объясняно. Учитель супер. Спасибо за ролик.

[@Georg.Löding]

I really like his explainations! 10 of 10 again. Thank you!

[@NadiehFan]

A different approach which does not require a substitution and which does not require having to deal with fractions starts by multiplying both sides by (x − 3)² to get rid of the fraction. We can then proceed as follows x²(x − 3)² + 9x² = 16(x − 3)² x²(x² − 6x + 9) + 9x² = 16(x − 3)² x²(x² − 6x + 18) = 16(x − 3)² (x² − 3(x − 3) + 3(x − 3))(x² − 3(x − 3) − 3(x − 3)) = 16(x − 3)² (x² − 3x + 9)² − 9(x − 3)² = 16(x − 3)² (x² − 3x + 9)² − 25(x − 3)² = 0 (x² − 3x + 9 + 5(x − 3))(x² − 3x + 9 − 5(x − 3)) = 0 (x² + 2x − 6)(x² − 8x + 24) = 0 x = −1 + √7 ⋁ x = −1 − √7 ⋁ x = 4 + 2i√2 ⋁ x = 4 − 2i√2

[@laurentbeyer2458]

I would have done same 👍🏻

[@TapanDas-ei3lw]

You're also a great genius ❤❤❤

[@anjalidilipogale4208]

More simpler

[@DanielDimov358]

Where does line №4 come from?

[@NadiehFan]

@@DanielDimov358 If you have two quantities a and b then their average is ½(a + b). Since a and b are equidistant from their own average, you can get the original quantities back by adding half their difference ½(a − b) to their average and by subtracting half their difference from their average. That is, we have a = ½(a + b) + ½(a − b) b = ½(a + b) − ½(a − b) The average of x² and x² − 6x + 18 is half their sum, which is x² − 3x + 9 = x² − 3(x − 3) and their difference is x² − (x² − 6x + 18) = 6x − 18 = 6(x − 3) so half their difference is 3(x − 3). Therefore, we have x² = x² − 3(x − 3) + 3(x − 3) and x² − 6x + 18 = x² − 3(x − 3) − 3(x − 3) Consequently, the product x²(x² − 6x + 18) can be written as (x² − 3(x − 3) + 3(x − 3))(x² − 3(x − 3) − 3(x − 3)) and applying the difference of two squares identity (a + b)(a − b) = a² − b² this gives (x² − 3(x − 3))² − 9(x − 3)² or (x² − 3x + 9)² − 9(x − 3)²

[@Miloudmiloud-z1m]

لغة الرياضيات لغة عالمية موحدة كقانون كرة القدم حقيقة ما اجمل الرياضيات مع الوضعيات المعقدة التي تبحر بك في اعماق التفكير لحلها كنت محبا لهذه المادة والحمد لله افادتني في الحياة

[@user-jj1sy3yh3v]

اكره مااكره الرياضيات وبسبب الرياضيات لم اصل لاهدافي وطموحي بالتعليم حسبي الله ونعم الوكيل

[@Sci-Marvels]

تماماً! لغة الرياضيات لغة عالمية توحدنا جميعاً، مثل قوانين كرة القدم. إنها تأخذنا في رحلة عميقة من التفكير لحل التحديات المعقدة. سعيد أن هذه المادة أفادتك في حياتك. الرياضيات حقاً رائعة! 😊📊

[@ajmiller7102]

Excellent. Straight to the solutions. Well done!

[@OlajiireTaiwo]

Please always make your solution simple, to the subject interesting to all, first clear the squares and solve within a minute

[@zineelabidinebouhaous9326]

حين كنا ندرس في الثانوي لا يمكن الانطلاق الا بمجموعة تعريف اي استثناء أن يكون x#3 ضرورية قبل الانطلاق

[@peterbyrne6394]

Thank you. very nice solved equation.

[@jahangirkhan1739]

If I watched a few videos like this, I would be mentally ill. 😂😂😂

[@annoint3d2mvu]

Me and you my friend, me and you! And the biggest question is when is this used in the real world?😅😅😅

[@milosspanka4256]

​@@annoint3d2mvumožno pri pristávaní rakety spoločmnosti X-space 😂 ale aj Albert Einstein je rád , že to nemusí počítať ... 😅😂😂

[@oxfordmathseries]

I prefer to get rid of fraction first, then solution will be of 2 minutes only

[@Ron_DeForest]

Would have been way faster and easier. So much less time to multiply through then gather terms.

[@agustinamariarivarolaberra3013]

Yes,I agree

[@jirah5397]

C'est tres long ca se fait dans 3ou4 ligne . Tu n as qu'à la donné à un Marocain

[@user-iz1wq7ou3r]

Или българин​@@jirah5397

[@AmiroVarinda]

😂❤​@@Ron_DeForest

[@nanakojo3711]

X x X + (3x3xXxX) - - - - - - - - - = 16 (XxX-3x3) > the Xs and 3s in the bracket crosses out. X x X = 16 X squared = 16 Now, remove the squared from the X, but to do that you have to put 16 in square root sign X = square root of 16 X = 4.

[@martinreichard3243]

I prefer to take the square root first and then expand with x-3. This is faster.

[@TapanDas-ei3lw]

Never ending marathon mathematics and brain 🧠 puzzled stories ever. Einstein newton and all superb mathematicians brain to be needed to understood not like me the stupid one..not getting any sweet juice extract from your mathematics, all energies i have lost to get it understood sir .really sir you've the genius brain and you're great.❤❤❤after this life i shall pray to God ,make me genius like you.❤❤❤❤❤

[@despoinavavoula2203]

This is crazy!

[@Aristotlearinzetv]

Binomial triangle can solve the equation in just few lines . You like extra sheet

[@ポン吉ポン田]

It can be solved by converting x to a trigonometric function. (x/4)^2 + (3x/(4(x-3)))^2 = 1 x/4 = cos 3x/(4(x-3)) = sin 16(cs)^2 - 18cs - 9 = 0

[@michaelhuppertz6738]

Nice solution specially how got the (x-3) out of 6x^3+18x^2, neat trick.

[@한효시]

ㅣ11

[@FelekeAsefa-p6y]

The correct answer is: $x = 0$ Explanation: In the previous responses, we found that the equation $x^{2}+(\frac {3x}{x-3})^{2}=16$ has only one solution, which is $x = 0$. This is because when we substitute $x = 0$ into the equation, both terms on the left-hand side become zero, resulting in the equation being satisfied. Therefore, $x = 0$ is the only value of x that satisfies the equation $x^{2}+(\frac {3x}{x-3})^{2}=16$.

[@edemkoffiaholou9808]

Pourquoi tu nous dis que racine carré de I est égal encore à i? Ça n'a pas de sens. Il y a un problème avec le résultat à la fin

[@josephissara3599]

So the underroot of negative one is i, how about the underroot of another negative number?

[@chimumaziku3186]

Yeeeeee my brain already bursted🤣🤣🤣

[@SsendaxAhmed]

😂

[@ednalee3083]

前面都看懂了，后面根号里公式不记得了，为什么-1开根号变成i😅

[@michaeltindan]

Why this long method?

[@esauogutu1702]

The secret to solving the equation is formulating it into a product of two quadratic expressions that gives zero

[@خالد-ح9ف-د2ل]

The solution become.complicated,rather than simplfied😮‬‏

[@gatabaziemmanuel]

Excellent

[@learncommunolizer]

Thank you so much 😀

[@PrudentVictor-q6d]

Good evening ma. Pls i need help in advance level math and physics pls.

[@bernayarimoglu5311]

In Turkiye, in university enterance examinations, you have only 1 or 2 minutes to solve this

[@YaoCaeuArmelKOUADIO]

You forget the square on i for the first solution 😅

[@HermanGATETE-nm5zj]

All these hours, just only to find X 😂

[@danielo.m.288]

Hope this can be useful in real life

[@gandhingira4071]

😂😂😂

[@sophyodago5062]

Where is 6× from

[@dewanfamuji5731]

You missed that x not allowed 3

[@劉河順-n7k]

It's a troubled solution.

[@romelitobragas9279]

Long methods of solution. What about short method sir?

[@mahmoudibra5822]

Beautiful

[@learncommunolizer]

Thank you very much

[@ASKstrive]

That is also how learning german will treat you as bigginer.

[@mintusaren895]

Why al zebra unknown vs unknown.

[@samsonhailu5133]

Waw 👍👍👍👍

[@lidzusakayina6842]

How did you know😮

[@gelanaadugna7808]

I am great sorry to you! why do you take 16 minute for 1 question? Why you make it coplex?

[@louiskah4753]

C'est une identité remarquable ❤

[@camairenentertainment254]

No wonder I was failing maths Did the teacher seriously expect me to memorise all this 😳

[@yanssala2214]

Parece equivalente a un polinomio de 4to grado, entonces habrian cuatro posibles valores de x X^4 -2X^3-6X^2+32X -16=0 es equivalente al problema propuesto. Un valor real obtenido por metodos iterativos seria 0,56895 con error no superior a. 0, 0006 😂😂😂

[@spelleg11]

Great.

[@RoluahpuiaColney]

This is a real mathemetics😮😮

[@nanakojo3711]

Its totally wrong. X =4

[@souzasilva5471]

Não acredito que o melhor professor do mundo tenha uma visão de primeira, para saber que o caminho é este aplicado nesta questão.(I don't believe that the best teacher in the world has the first-class vision to know which way is applied to this issue.)

[@bedelludesta5508]

I don’t understand it. First you need to explain what are 2 and x represent

[@mbuyaelyaoni6266]

Mmh, till tomorrow 😅😅

[@TewabechEshete-pi3fz]

too long solution that's why many of us ignored Maths . From your side, excellent potential !

[@ahadjonxamidov8940]

Zo‘r

[@hope9203]

Very nice ✅

[@learncommunolizer]

Thanks ✌️

[@user-io7ul8em1o]

We need to know the value of x

[@olympiadeduworks]

great

[@jahangirkhan1739]

Why does algebra have such complications and confusing sequence to solve the problem?😂😂

[@samswift4921]

C 1.4 hrs or 84 minutes.

[@ahmedder3608]

Solution lente

[@user-io7ul8em1o]

This is so long You could finish this after 3 or 4 lines

[@JuanCarlosMontoya-z2k]

Para que sirve eso, si no sabe para que es

[@joachimkaran6687]

Amna kitu umetupereka op

[@FathimaHararyy]

Thank you so much 😂😂😂😂😂😂😂😂

[@learncommunolizer]

Welcome 😊

[@gatabaziemmanuel]

Good

[@learncommunolizer]

Thanks

[@SaraAlam-l9t]

It's so simple

[@mohamedsalah5525]

4-

[@alexpangani1944]

Nice

[@learncommunolizer]

Thanks

[@ЗемфираШаргазина]

Почему в ответе Вы сократили на 2..???? Так нельзя!!!

[@olympiadeduworks]

nice

[@serifahmetgumus5763]

Kafa ütülemek matematiği hiç sevmem

[@NadiehFan]

Yet another method is to note that (x − 3)² = x² − 6x + 9 and therefore x² = (x − 3)² + 6x − 9 so we can rewrite the equation x² + (3x/(x − 3))² = 16 as (x − 3)² + 6x − 9 + (3x/(x − 3))² = 16 or (x − 3)² + 6x + (3x/(x − 3))² − 25 = 0 Here (x − 3)² + 6x + (3x/(x − 3))² is a perfect square because 6x is twice the product of (x − 3) and 3x/(x − 3) and we also have 25 = 5² so we get ((x − 3) + 3x/(x − 3))² − 5² = 0 which gives us a difference of two squares at the left hand side whereas the right hand side is zero. This allows us to factor the left hand side using the difference of two squares identity and then apply the zero product property. However, before doing so we first multiply both sides by (x − 3)² to eliminate the fraction so we get ((x − 3)² + 3x)² − (5(x − 3))² = 0 (x² − 3x + 9)² − (5x − 15)² = 0 (x² + 2x − 6)(x² − 8x + 24) = 0 x = −1 + √7 ⋁ x = −1 − √7 ⋁ x = 4 + 2i√2 ⋁ x = 4 − 2i√2

[@FatimaFatima-eu7ix]

Mamma mia 😢

[@rajanisarmah6307]

Aare vaai itni si sota math aur kitna lomba korega??

[@ВасильГецко-з7ш]

ТА ЧТО ЗДЕСЬ РИШАТЬ,ЛИХКО И ПРОСТО ,КАК НА ГОРШКЕ ПОСИДЕТЬ🪺

[@MinhTran-xy7hy]

X2( x3)2=16X yes and nos (x-3). =^ ./.

[@francescobonacorsi5778]

Impressionante

[@TheTeller2012]

X-x/3-1=0

[@AVELINETOMEKA]

Make people understand 😅😅😅😅😅😅😅😅😅

[@InkoNu-nv9tf]

=2

[@LANNGUYEN-kv6jl]

The ưay you solve this equation sis SO LONG.

[@jummyshaki2922]

So long a letter

[@user-io7ul8em1o]

Long way

[@Dino-Kukolj]

Too long explanation and you should have switched to the quadratic equasion earlier, it was inevitable and as far as I know there is no negative square root, because any negative number squared gives a positive number, and a positive number squared is anyway positive

[@ZoukalneLeonLaboulbe]

Très longue comme méthode

[@lenchorivera8543]

Me estoy acordando de mis clases en la prepa…

[@joachimkaran6687]

Umekosea APO

[@mintusaren895]

Result same

[@mohammadwatan1992]

👍

[@learncommunolizer]

👍👍👍👍

[@fentlove9392]

This shows us how long, hard and tiresome it is to come up with a constitution..Dont blame your government too much😅

[@BambadjiFall]

❤❤❤❤❤❤❤❤❤❤

[@DibweKalambay-cu6ts]

I find X=4

[@tamarshahverdyan2723]

# 30 #

[@АЛЕКСАНДРВАН-о9ч]

Ой, раздули из простого решения целое математическое расследование

[@tatyanasnisarenko1627]

Х=2

[@Alexanderdegrey765Majuedkelucb]

Mbledh e zbret ate q t jep mendja pa terrible fikse

[@DimosoAbduly]

4

[@gandhingira4071]

Jesus!

[@michaeltindan]

The method you're using is unnecessarily tooooo loooong. This is a simple question that requires a simple method.

[@ahmadikininga239]

Please give a sense to those x and numbers. Be realistic. Don't speak in signs. Unless you are hiding something.