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

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  • Опубліковано 29 жов 2024

КОМЕНТАРІ • 46

  • @Segalmed
    @Segalmed 2 місяці тому +15

    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 місяці тому

      A nice way of elaboration

    • @esp-l1n
      @esp-l1n Місяць тому

      That's exactly what I did

    • @dr_Sehaq
      @dr_Sehaq Місяць тому

      Not a math guy. But good explanation

    • @Sci-Marvels
      @Sci-Marvels Місяць тому

      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 місяці тому +1

    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 Місяць тому

    Génial ! J'étais bien partie, mais j'oubliais qu'on pouvait faire un changement de variable !..pour simplifier le calcul

  • @bpark10001
    @bpark10001 2 місяці тому +1

    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 Місяць тому

      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 місяці тому +2

    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 місяці тому

    Excelente. Gracias

  • @richi2casa127
    @richi2casa127 Місяць тому

    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 місяці тому +2

    here is another one - s^2 + 5x -6 is factorable - ( x +6) (x -1 ) - quadratic equation not needed.

    • @Sci-Marvels
      @Sci-Marvels Місяць тому

      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 місяці тому

    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 місяці тому +1

    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 Місяць тому

    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 Місяць тому +1

    1

  • @williamBryan-k2e
    @williamBryan-k2e 2 місяці тому

    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 Місяць тому

    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 Місяць тому

      Отличное разложение задачи! Вы чётко показали, как перейти от замены переменной к решению уравнений и нахождению корней. Прекрасная работа!

  • @hakanerci4372
    @hakanerci4372 2 місяці тому

    X=-6
    (x+1).(x+2).(x+3).(x+4)=120

  • @稲次将人
    @稲次将人 2 місяці тому

    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 місяці тому

    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 Місяць тому

      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 Місяць тому

    -6

  • @สมเจียวเจริญโสภา

    ตอบ X=1

  • @sankariraman6453
    @sankariraman6453 2 місяці тому

    X=1

  • @lioneljohnpilcher204
    @lioneljohnpilcher204 2 місяці тому +1

    I don't have first one. Please help thanks ❤ l mean it please.

  • @TeachChulbul
    @TeachChulbul Місяць тому

    putting the value of 1 =x

  • @TeachChulbul
    @TeachChulbul Місяць тому

    X=1then multiply

  • @JeromeTui
    @JeromeTui 2 місяці тому

    1:20

  • @YoussoufSOULAIMANA
    @YoussoufSOULAIMANA Місяць тому

    X = 1 is the answer

  • @ahmetmete8366
    @ahmetmete8366 2 місяці тому

    Böyle matematik düşman başına. O ne öyle ortaokul çocuğuna anlatırmış gibi.

    • @Sci-Marvels
      @Sci-Marvels Місяць тому

      Matematik bazen karmaşık görünebilir, ama basit bir şekilde anlatıldığında herkesin anlaması daha kolay oluyor. Herkesin seviyesi farklı olabilir, ama amaç hepimizin öğrenmesi. 😊

  • @SuperAnangs
    @SuperAnangs Місяць тому

    1 sec, x=1

  • @haitrieu0051
    @haitrieu0051 2 місяці тому

    a+b+c=0, x=1 ,x=c/a , x=1 ; x=-6

    • @lioneljohnpilcher204
      @lioneljohnpilcher204 2 місяці тому

      Incorrect. A vector can amount it's velocity. I need someone to look just pass proxima B please

    • @AmédéeGerest
      @AmédéeGerest 2 місяці тому +3

      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 місяці тому +1

      ​@@AmédéeGerestyes

  • @822Adi
    @822Adi Місяць тому

    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 Місяць тому

    1

  • @ramakrishnanwarrier6662
    @ramakrishnanwarrier6662 Місяць тому

    X= 1

  • @lakshmikutty2255
    @lakshmikutty2255 2 місяці тому

    x=1

  • @sakirhussain665
    @sakirhussain665 2 місяці тому

    X=1

  • @yordanyordanov7257
    @yordanyordanov7257 Місяць тому

    x=1