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  1. #1
    emrldz
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    Algebra Help

    Cubic Equations

    Q -
    (i) Find the three linear factors of x + 5x - 2x - 24.
    (ii) Hence, or otherwise, solve the equation x + 5x - 2x - 24 = 0

    Please supply a formula aswell so I can solve these in the future! Thankyou




    PLEASE HELP! VERY VERY URGENT
    Last edited by emrldz; November 19th 2007 at 10:31 AM.
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by emrldz View Post
    Q -
    (i) Find the three linear factors of x + 5x - 2x - 24.
    (ii) Hence, or otherwise, solve the equation x + 5x - 2x - 24 = 0

    Please supply a formula aswell so I can solve these in the future! Thankyou




    PLEASE HELP! VERY VERY URGENT
    Hello,

    the good news first: There exists a formula to solve cubic equation: Cardano's formula. It is very complicated and therefore it is very easy to make a lot of silly mistakes.

    I can only sho you how I would do your problem:
    1. The question indicates that there are rational solutions.
    2. Try to find one solution. It must be a factor of 24. I've found x = 2 is a solution because:
    2^3+5 \cdot 2^2 - 2 \cdot 2 - 24 = 8+20-4-24 = 0
    3. Now use synthetic division:
    Code:
          |  1    5     -2    -24          <---all coefficients
          |  0    2     14     24
    ------|-------------------------
    x = 2 |  1   7      12      0         <---next guess: x = -3
          |  0   -3     -12
    ------|-------------------------
    x = -3| 1    4      0
    From the last row you know x + 4 = 0~\iff~x = -4

    Therefore: x^3+5x^2-2x-24 = (x-2)(x+3)(x+4)

    A product equals zero if one of the factors equal zero:

    x^3+5x^2-2x-24 = =~\iff~x-2=0~\vee~x+3=0~\vee~x+4=0
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