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Math Help - hard equation !

  1. #1
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    hard equation !

    HI !!

    How can I solve this equation ???


    (x+1)(x+2)(x+3)(x+4) = 120 \ \ \ \ \ \
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  2. #2
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    1. Expand

    2. Set equal to 0

    3. Factorise if possible and solve using NFL.
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  3. #3
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    Here's a clever little shortcut. 120 is equal to 5!. So x = 1 works.
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  4. #4
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    Quote Originally Posted by yehoram View Post
    HI !!

    How can I solve this equation ???


    (x+1)(x+2)(x+3)(x+4) = 120 \ \ \ \ \ \
    Dear yehoram,

    (x+1)(x+2)(x+3)(x+4) = 120=2\times 3\times 4\times 5

    Therefore it is clear that x=1 is a solution.

    (x+1)(x+2)(x+3)(x+4)-120=(x-1)(Ax^3+Bx^2+Cx+D)=0

    Evaluating A,B,C and D will give you a cubic equation. The solutions can be obtained using several methods such as the Cardano's method. Refer: Cubic function - Wikipedia, the free encyclopedia
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  5. #5
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    The key to this problem is to partition the products into two : (1) the first and the last , (2) the 2nd and the 3rd , ie

     (x+1)(x+2)(x+3)(x+4) = 120

     [(x+1)(x+4)][(x+2)(x+3)] = 120

     (x^2 + 5x + 4 )(x^2 + 5x + 6) = 120

    Then sub.  u=  x^2 + 5x + 5 so we have

     (u-1)(u+1) = 120 or

     u^2 = 121 thus

     u = 11 or  u = -11

     x^2 + 5x - 6 = 0 or  x^2 + 5x + 16 = 0

    The four solutions are

     -6,1 , \frac{-5+\sqrt{39} i }{2} , \frac{-5-\sqrt{39} i }{2}
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  6. #6
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    Hello, yehoram!

    How can I solve this equation?

    . . (x+1)(x+2)(x+3)(x+4) \:=\: 120

    We have: the product of four "consecutive" numbers equals 120.

    The only candidates are: . \{2,3,4,5\}\,\text{ and }\,\{\text{-}5,\text{-}4,\text{-}3,\text{-}2\}

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  7. #7
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    Quote Originally Posted by yehoram View Post
    HI !!

    How can I solve this equation ???


    (x+1)(x+2)(x+3)(x+4) = 120 \ \ \ \ \ \
    Along the same lines as Dr Steve's "zwischenzug", what if x=-6\;\;?

    (my answer looks a bit silly now. What's with the clock? Soroban's response wasn't there when I started!)
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  8. #8
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    Quote Originally Posted by yehoram View Post
    (x+1)(x+2)(x+3)(x+4) = 120 \ \ \ \ \ \
    x^4 + something + 24 = 120
    x^4 = 96 - something
    x^4 < 96

    Well?!
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