China | Can you solve this ? | A Nice Math Olympiad Algebra Problem

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Learncommunolizer

Learncommunolizer

Күн бұрын

Пікірлер: 46
@Segalmed
@Segalmed 2 ай бұрын
120 = 5! = %*4*3*2*1 = 5*4*3*2 => the trivial solution is x =1. But there must be 4 more solutions since this is a 4th degree polynomial. Another one is obviously -6 since there are four brackets, so if all are negative the result is still positive and we get the same result, if one bracket is -5 the next -4 and so on. My guess is that the the remaining two solutions are complex. So, divide the whole polynomial by (x+6) and (x-1) to get a quadratic equation, that can be solved by the usual formula. Edit: That equation is x^2 + 5x +16 which has the complex solutions -2.5 +/- i*0.5*SQRT(39).
@ManojkantSamal
@ManojkantSamal 2 ай бұрын
A nice way of elaboration
@esp-l1n
@esp-l1n Ай бұрын
That's exactly what I did
@dr_Sehaq
@dr_Sehaq Ай бұрын
Not a math guy. But good explanation
@Sci-Marvels
@Sci-Marvels Ай бұрын
Great breakdown! You've clearly worked through the polynomial to find all the solutions, including the complex ones. I appreciate the detailed explanation and the logical steps you took to solve it. Well done!
@iitiansaurabhbhu
@iitiansaurabhbhu 2 ай бұрын
2×3×4×5=120 from here by hit and trial method......... X+4=5(bigger with bigger) X=1 and -5×-4×-3×-2 X+1=-5(smaller with smaller) X=-6 Your method is correct but for saving time i analyzed the pattern and hence!!!!!!!!!!! Your method is also appreciable🎉
@pierrettebalazut9407
@pierrettebalazut9407 Ай бұрын
Génial ! J'étais bien partie, mais j'oubliais qu'on pouvait faire un changement de variable !..pour simplifier le calcul
@bpark10001
@bpark10001 2 ай бұрын
One look at 120, it is 5 factorial. The equation has ascending numbers in it, so X = 1 can be determined by inspection. Because there is even number of factors, negative numbers can give positive result so -6 can also be determined by inspection. These 2 factors (X - 1 & X + 6) can be factored out to give the quadratic equation X² + 5X + 16 = 0. The roots of this are (-5 + i√39)/2 and (-5 - i√39)/2. These are not real numbers so are not part of problem solution, but are the other 2 roots of the 4th order equation for completeness.
@Sci-Marvels
@Sci-Marvels Ай бұрын
Nice observation! Spotting the factorial and using inspection to quickly find the real roots is a great approach. And you’re right about the complex roots-they complete the solution even if they aren’t part of the original problem’s real-number solution. Thanks for sharing your method! 👏😊
@vacuumcarexpo
@vacuumcarexpo 2 ай бұрын
Using x(x-a)(x-2a)(x-3a)+a^4=(x(x-3a)+a^2)^2, (x-1)(x-4)+1=±11
@cachotrelles4715
@cachotrelles4715 2 ай бұрын
Excelente. Gracias
@richi2casa127
@richi2casa127 Ай бұрын
import numpy as np # Coefficients of the quartic equation coefficients = [1, 10, 35, 50, -96] # Solve the equation roots = np.roots(coefficients) roots
@williamBryan-k2e
@williamBryan-k2e 2 ай бұрын
here is another one - s^2 + 5x -6 is factorable - ( x +6) (x -1 ) - quadratic equation not needed.
@Sci-Marvels
@Sci-Marvels Ай бұрын
Good point! When the quadratic is factorable like this, it's definitely quicker to just factor it directly. No need for the quadratic formula when the factors are right there! Thanks for the reminder. 😊👍
@skverma7278
@skverma7278 2 ай бұрын
Sir, we have to find 4 factors of 120 such, these are consecutive numbers. Factors of 120: 2×2×2×3×5 Possible arrangement is 2×3×4×5 Hence value of x is 1. This will be true for negative values of factors i.e. for -2,-3,-4,& -5 for which x will be -6.
@walterwen2975
@walterwen2975 2 ай бұрын
Math Olympiad: (x + 1)(x + 2)(x + 3)(x + 4) = 120, x ϵ R; x =? First method: (x + 1)(x + 2)(x + 3)(x + 4) = 120 = (2)(3)(4)(5) = (1 + 1)(1 + 2)(1 + 3)(1 + 4) = (- 6 + 1)(- 6 + 2)(- 6 + 3)(- 6 + 4) = (- 5)(- 4)(- 3)(- 2); x = 1 or x = - 6 Missing two complex value roots Second method: [(x + 2)(x + 3)][(x + 1)(x + 4)] = (x² + 5x + 6)(x² + 5x + 4) = 0 (x² + 5x + 6)(x² + 5x + 6 - 2) = (x² + 5x + 6)² - 2(x² + 5x + 6) = 120 (x² + 5x + 6)² - 2(x² + 5x + 6) - 120 = (x² + 5x + 6 - 12)(x² + 5x + 6 + 10) = 0 (x² + 5x - 6)(x² + 5x + 16) = (x - 1)(x + 6)(x² + 5x + 16) = 0 x² + 5x + 16 = 0, x = (- 5 ± i√39)/2; Rejected, complex value roots x - 1 = 0; x = 1 or x + 6 = 0; x = - 6 Answer check: (x + 1)(x + 2)(x + 3)(x + 4) = 120; Confirmed as shown in First method Final answer: x = 1 or x = - 6
@AlfredoMirandaSanchez
@AlfredoMirandaSanchez Ай бұрын
I've mentally solved this looking for two factors of 120 that at the same time had two factors which difference is one.: (2x3)x(5x4)=120. X=1 is the value that match the equation.
@nadiapianist
@nadiapianist Ай бұрын
1
@williamBryan-k2e
@williamBryan-k2e 2 ай бұрын
I got x =1 from trial and error before seeing the snaswer.. but as someone said - should be 4 solutions as 4th order poly. so I will watch
@Евгений-ю2р
@Евгений-ю2р Ай бұрын
Напишу кратко (x^2+5x+3+1)(x^2+5x+3+3)=120 обозначим (x^2+5x+3)- b тогда имеем (b+1)(b+3)=120 далее b^2+4b-117=0 отсюда b1=9 ,а b2=-13 т.е x^2+5x+3=9 и x^2+5x+3=-13 как видим x^2+5x-6=0 и второе x^2+5x+16=0 далее всё остальное подробно на видео.Ответ: x1=1; x2=-6
@Sci-Marvels
@Sci-Marvels Ай бұрын
Отличное разложение задачи! Вы чётко показали, как перейти от замены переменной к решению уравнений и нахождению корней. Прекрасная работа!
@hakanerci4372
@hakanerci4372 2 ай бұрын
X=-6 (x+1).(x+2).(x+3).(x+4)=120
@稲次将人
@稲次将人 2 ай бұрын
2・3・4・5=120だからx=1 x=-6のとき(-5)(-4)(-3)(-2)=120 ∴x=1,-6
@m.s.9023
@m.s.9023 2 ай бұрын
Using a(a+b)(a+2b)(a+3b)+b^4 = (a^2+3ab+b^2)^2, {(x+1)^2 + 3(x+1)+1}^2 = 11^2
@Sci-Marvels
@Sci-Marvels Ай бұрын
Impressive use of identities! It’s always fascinating to see such elegant algebraic manipulations that simplify the problem so effectively. Thanks for sharing this approach! 😊👏
@manjeetjoon7
@manjeetjoon7 Ай бұрын
-6
@สมเจียวเจริญโสภา
@สมเจียวเจริญโสภา Ай бұрын
ตอบ X=1
@sankariraman6453
@sankariraman6453 2 ай бұрын
X=1
@lioneljohnpilcher204
@lioneljohnpilcher204 2 ай бұрын
I don't have first one. Please help thanks ❤ l mean it please.
@TeachChulbul
@TeachChulbul Ай бұрын
putting the value of 1 =x
@TeachChulbul
@TeachChulbul Ай бұрын
X=1then multiply
@JeromeTui
@JeromeTui 2 ай бұрын
1:20
@YoussoufSOULAIMANA
@YoussoufSOULAIMANA Ай бұрын
X = 1 is the answer
@ahmetmete8366
@ahmetmete8366 2 ай бұрын
Böyle matematik düşman başına. O ne öyle ortaokul çocuğuna anlatırmış gibi.
@Sci-Marvels
@Sci-Marvels Ай бұрын
Matematik bazen karmaşık görünebilir, ama basit bir şekilde anlatıldığında herkesin anlaması daha kolay oluyor. Herkesin seviyesi farklı olabilir, ama amaç hepimizin öğrenmesi. 😊
@SuperAnangs
@SuperAnangs Ай бұрын
1 sec, x=1
@haitrieu0051
@haitrieu0051 2 ай бұрын
a+b+c=0, x=1 ,x=c/a , x=1 ; x=-6
@lioneljohnpilcher204
@lioneljohnpilcher204 2 ай бұрын
Incorrect. A vector can amount it's velocity. I need someone to look just pass proxima B please
@AmédéeGerest
@AmédéeGerest 2 ай бұрын
I think I found a quicker way this time : y = x“ + 5x + 4 y(y+2) = 10.12 y = 10 x" + 5x + 4 = 10 x" + 5x - 6 = 0 (x-1)(x+6) = 0 S = {-6;1} Is this correct?
@iitiansaurabhbhu
@iitiansaurabhbhu 2 ай бұрын
​@@AmédéeGerestyes
@822Adi
@822Adi Ай бұрын
Such a fool You can simply factorise 120 so you get 2 3 4 5 continuous numbers so in order to get those simply add 1 So x = 1
@neeldesilva8088
@neeldesilva8088 Ай бұрын
1
@ramakrishnanwarrier6662
@ramakrishnanwarrier6662 Ай бұрын
X= 1
@lakshmikutty2255
@lakshmikutty2255 2 ай бұрын
x=1
@sakirhussain665
@sakirhussain665 2 ай бұрын
X=1
@yordanyordanov7257
@yordanyordanov7257 Ай бұрын
x=1
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