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Math Help - Urgent help needed on equation!!!

  1. #1
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    Exclamation Urgent help needed on equation!!!

    Hey, i have to complete an AS level core one maths exam paper but i am completely stuck on the following question: Solve the equation x6+26x3-27=0

    I have nooooo idea what to do! If someone could please answer the question with a full solution so i can understand it i would be most grateful!!!

    Thank you very much!
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  2. #2
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    Hello, LoveDeathCab!

    Solve: .x^6 + 26x^3 - 27 = 0
    Sure, you know what to do . . . try to factor it . . .


    We have: .(x - 1)(x + 27) .= .0

    Then solve the two equations:

    . . x - 1 .= .0 . . x = 1 . . x = 1

    . . x + 27 .= .0 . . x = -27 . . x = -3

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  3. #3
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    Oh my goodness!

    You are a genius! Thank you sooooo much! It was doing my head in! I feel rather stupid now! Haha thanks!
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  4. #4
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    Quote Originally Posted by LoveDeathCab View Post
    Hey, i have to complete an AS level core one maths exam paper but i am completely stuck on the following question: Solve the equation x6+26x3-27=0

    I have nooooo idea what to do! If someone could please answer the question with a full solution so i can understand it i would be most grateful!!!

    Thank you very much!
    Quote Originally Posted by Soroban View Post
    Hello, LoveDeathCab!

    Sure, you know what to do . . . try to factor it . . .


    We have: .(x - 1)(x + 27) .= .0

    Then solve the two equations:

    . . x - 1 .= .0 . . x = 1 . . x = 1

    . . x + 27 .= .0 . . x = -27 . . x = -3

    I am surprised that Soroban did not mention this: you are missing 4 complex solutions here. (Perhaps the "AS level core exam" means something to him. It doesn't to me!) If you want them, this is how to get them:

    x^6 + 26x^3 - 27 = 0

    (x^3 - 1)(x^3 + 27) = 0

    Now,
    x^3 - 1 = (x - 1)(x^2 + x + 1)
    x^3 + 27 = (x + 3)(x^2 - 3x + 9)

    So:
    x^6 + 26x^3 - 27 = (x^3 - 1)(x^3 + 27) = (x - 1)(x^2 + x + 1)(x + 3)(x^2 - 3x + 9) = 0

    To get the solutions set each individual factor equal to 0.

    I get
    x = 3/2 (+/-) I*(3/2)*sqrt{3}, (-1/2) (+/-) I*(1/2)sqrt{3}, 1, -3
    where I^2 = -1.

    -Dan
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