Can you Pass Pure Mathematics Entrance Exam from Oxford University ?

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  • Опубликовано: 30 окт 2024

Комментарии • 9

  • @kareolaussen819
    @kareolaussen819 День назад +4

    First search for integer solutions. Since none is found try factorization into two quadratic terms with integer coefficients,
    (x^2 + px + q)(x^2 - px + r) =
    x^4 + (q+r-p^2)x^2 - (q-r)px + qr =
    x^4 -12x -5.
    We must have qr = -5 with q+r positive (equal to p^2). The only integer solution is q=5, r=-1, which gives p^2=4. This matches the condition that (q-r)p=12 when p=2. Hence,
    (x^2 + 2x +5)(x^2 -2x-1) = x^4 -12x -5.

  • @ShriH-d1o
    @ShriH-d1o 14 часов назад +1

    0:59 Let X^4 - 12X - 5 =(X^2 +aX+ p)(X^2 +bX + q) so
    a+b=0 ; pq= -5;
    p + q + ab = 0; aq + pb = - 12
    solving a = 2; b = - 2 ; p = 5; q = - 1 ...

  • @jorgevilas1603
    @jorgevilas1603 2 дня назад +2

    Beautiful exercise and very well resolved, thank you.

    • @superacademy247
      @superacademy247  2 дня назад

      Thanks for watching! 🙏😊I’m glad you found it helpful! 💯💖

  • @key_board_x
    @key_board_x 2 дня назад +1

    x⁴ - 12x - 5 = 0 ← it would be interesting to have 2 squares on the left side (because power 4)
    Let's rewrite the equation by introducing a variable "λ" which, if carefully chosen, will produce a square on the left side
    Let's tinker a bit with x⁴ as the beginning of a square: x⁴ = (x² + λ)² - 2λx² - λ²
    x⁴ - 12x - 5 = 0 → where: x⁴ = (x² + λ)² - 2λx² - λ²
    (x² + λ)² - 2λx² - λ² - 12x - 5 = 0
    (x² + λ)² - [2λx² + λ² + 12x + 5] = 0 → let's try to get a second member as a square
    (x² + λ)² - [(2λ).x² + 12x + (λ² + 5)] = 0 → a square into […] means that Δ = 0 → let's calculate Δ
    Δ = 12² - 4.[2λ * (λ² + 5)] → then, Δ = 0
    12² - 8λ.(λ² + 5) = 0
    8λ.(λ² + 5) = 12²
    λ.(λ² + 5) = 18 → we can observ an obvious solution
    λ = 2
    Restart
    (x² + λ)² - [(2λ).x² + 12x + (λ² + 5)] = 0 → where: λ = 2
    (x² + 2)² - [4x² + 12x + 9] = 0 ← we can see a square inside […]
    (x² + 2)² - (2x + 3)² = 0 → a² - b² = (a + b).(a - b)
    [(x² + 2) + (2x + 3)].[(x² + 2) - (2x + 3)] = 0
    (x² + 2 + 2x + 3).(x² + 2 - 2x - 3) = 0
    (x² + 2x + 5).(x² - 2x - 1) = 0
    First case: (x² + 2x + 5) = 0
    x² + 2x + 5 = 0
    Δ = 2² - (4 * 5) = 4 - 20 = - 16 = 16i²
    x = (- 2 ± 4i)/2
    → x = - 1 ± 2i
    Second case: (x² - 2x - 1) = 0
    x² - 2x - 1 = 0
    Δ = (- 2)² - (4 * - 1) = 4 + 4 = 8
    x = (2 ± √8)/2
    x = (2 ± 2√2)/2
    → x = 1 ± √2

  • @9허공
    @9허공 День назад

    Since there is no x^3, x^2 terms in the given equation,
    x^4 - 12x - 5 = (x^2 + a/2)^2 - a(x + b/2)^2 = x^4 - abx + (a^2 - ab^2)/4
    => -ab = -12 & (a^2 - ab^2)/4 = -5 => a^2 - ab^2 = a^2 - a(12/a)^2 = -20
    => a^3 + 20a - 144 = (a - 4)(a^2 + 4a + 36) = 0 => a = 4 => b = 3
    => x^4 - 12x - 5 = (x^2 + 2)^2 - 4(x + 3/2)^2 = (x^2 + 2)^2 - (2x + 3)^2 = (x^2 + 2x + 5)(x^2 -2x - 1)

  • @RealQinnMalloryu4
    @RealQinnMalloryu4 2 дня назад

    (x^4)^2 ➖( 12)^2 ➖ 5={x^16 ➖ 144} ➖ 5=128 ➖ (5)^2={128 ➖ 25}=103 10^10^3.2^5^2^5^3 1^1^2^1^3.23 (x ➖ 3x+2).

  • @sorinescu123456789
    @sorinescu123456789 День назад

    Taking super small steps and writing every single detail and repeating the writing of everything makes it boring and unnerving to the highest degree. I'd rather give up math altogether than see another RUclips clip by you. Not that you weren't correct. Just maddening...

    • @superacademy247
      @superacademy247  День назад +1

      When I move with speed avoiding trivial steps some viewers complain I'm too fast for them to understand . I think I need to strike a balance of all my viewership.