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.
@@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
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
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! 😊📊
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.
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 $x^{2}+(\frac {3x}{x-3})^{2}=16$.
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.❤❤❤❤❤
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.)
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 😂😂😂
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
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
![Blox Fruits Dragon Rework Update [Full Stream]](http://i.ytimg.com/vi/EqDAp8udhm0/mqdefault.jpg)
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.
Di
..
Di
..
I did this kind of maths about 40 years ago. Thanks for the reminder son.
May orpy
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.
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
I would have done same 👍🏻
You're also a great genius ❤❤❤
More simpler
Where does line №4 come from?
@@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
Excellent. Straight to the solutions. Well done!
If I watched a few videos like this, I would be mentally ill. 😂😂😂
Me and you my friend, me and you!
And the biggest question is when is this used in the real world?😅😅😅
@@annoint3d2mvumožno pri pristávaní rakety spoločmnosti X-space 😂 ale aj Albert Einstein je rád , že to nemusí počítať ... 😅😂😂
Please always make your solution simple, to the subject interesting to all, first clear the squares and solve within a minute
I really like his explainations! 10 of 10 again. Thank you!
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
اكره مااكره الرياضيات وبسبب الرياضيات لم اصل لاهدافي وطموحي بالتعليم حسبي الله ونعم الوكيل
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! 😊📊
حين كنا ندرس في الثانوي لا يمكن الانطلاق الا بمجموعة تعريف اي استثناء أن يكون x#3 ضرورية قبل الانطلاق
Thank you. very nice solved equation.
Как прекрасен язык математики. Всё понятно, подробно объясняно. Учитель супер. Спасибо за ролик.
I prefer to take the square root first and then expand with x-3. This is faster.
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.
In fact I wonder how in this life am I going to use all this effort to solve my problems 🤷🏿♂️😂😂😂
Binomial triangle can solve the equation in just few lines . You like extra sheet
I prefer to get rid of fraction first, then solution will be of 2 minutes only
Would have been way faster and easier. So much less time to multiply through then gather terms.
Yes,I agree
C'est tres long ca se fait dans 3ou4 ligne . Tu n as qu'à la donné à un Marocain
Или българин@@jirah5397
😂❤@@Ron_DeForest
The solution become.complicated,rather than simplfied😮
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 $x^{2}+(\frac {3x}{x-3})^{2}=16$.
what about if you can make both side in square root and find the value of x=12 andx=4
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
The secret to solving the equation is formulating it into a product of two quadratic expressions that gives zero
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.❤❤❤❤❤
In Turkiye, in university enterance examinations, you have only 1 or 2 minutes to solve this
Yeeeeee my brain already bursted🤣🤣🤣
😂
I need to learn more from him
You forget the square on i for the first solution 😅
So the underroot of negative one is i, how about the underroot of another negative number?
All these hours, just only to find X 😂
No wonder I was failing maths
Did the teacher seriously expect me to memorise all this 😳
Hope this can be useful in real life
😂😂😂
Nice solution specially how got the (x-3) out of 6x^3+18x^2, neat trick.
ㅣ11
It's a troubled solution.
You missed that x not allowed 3
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.)
Bro just overcomplicated the whole thing🤦🏾♀️
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
Why this long method?
too long solution that's why many of us ignored Maths . From your side, excellent potential !
Thanks for the vidéo .
You're welcome 🙏❤️🙏
This is crazy!
Good evening ma. Pls i need help in advance level math and physics pls.
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 😂😂😂
How many marks om this question
Where is 6× from
I am great sorry to you! why do you take 16 minute for 1 question? Why you make it coplex?
Why al zebra unknown vs unknown.
That is also how learning german will treat you as bigginer.
This is a real mathemetics😮😮
How did you know😮
Long methods of solution.
What about short method sir?
I don’t understand it. First you need to explain what are 2 and x represent
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😅
Its totally wrong. X =4
Beautiful
Thank you very much
We need to know the value of x
I like his solutions
Excellent
Thank you so much 😀
Mmh, till tomorrow 😅😅
Zo‘r
C'est une identité remarquable ❤
Why does algebra have such complications and confusing sequence to solve the problem?😂😂
This is so long
You could finish this after 3 or 4 lines
Very nice ✅
Thanks ✌️
Почему в ответе Вы сократили на 2..????
Так нельзя!!!
Waw 👍👍👍👍
Thanks 😊
thank
The ưay you solve this equation sis SO LONG.
C 1.4 hrs or 84 minutes.
Solution lente
Thank you so much 😂😂😂😂😂😂😂😂
Welcome 😊
Amna kitu umetupereka op
Kafa ütülemek matematiği hiç sevmem
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
Para que sirve eso, si no sabe para que es
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
Great.
X2( x3)2=16X yes and nos
(x-3). =^ ./.
ТА ЧТО ЗДЕСЬ РИШАТЬ,ЛИХКО И ПРОСТО ,КАК НА ГОРШКЕ ПОСИДЕТЬ🪺
I am confused. .but thanks
It's so simple
Aare vaai itni si sota math aur kitna lomba korega??
great
Make people understand 😅😅😅😅😅😅😅😅😅
Ой, раздули из простого решения целое математическое расследование
So long a letter
Please give a sense to those x and numbers. Be realistic. Don't speak in signs. Unless you are hiding something.
The method you're using is unnecessarily tooooo loooong. This is a simple question that requires a simple method.
Good
Thanks
This shows us how long, hard and tiresome it is to come up with a constitution..Dont blame your government too much😅
3+√23
Nice
Thanks
Result same
X-x/3-1=0
My brain just turned to mush after watching this🫠
Clumsy solution. You try to over complicate the solution.
I find
X=4
Faluche its wrong
nice
Très longue comme méthode
Too long way method.
mali yong sulution, Pinaha ba mali naman