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Quadratic Equations: Finding three consecutive positive integers given sum of squares

  1. Oct 20, 2012 #1
    1. Three consecutive positive integers are such that the sum of the squares of the first two and the product of the other two is 46. Find the numbers. Variables: x. Three numbers: (x), (x + 1), (x + 2)



    2. (I think, although I'm not sure.) x2 + (x + 1)2 + (x + 1)(x + 2) = 46



    3.
    x2 + (x + 1)2 + (x + 1)(x + 2) = 46
    ⇒ x2 + x2 + 1 + 2x + x2 + 3x + 2 = 46
    ⇒ 3x2 + 5x + 3 = 46
    ⇒ 3x2 + 5x - 43 = 0
    ⇒ x2 + (5/3)x - 43/3 = 0/3
    ⇒ x2 + (2)(5/6)(x) + (5/3)2 - (5/3)2 - 43/3 = 0
    ⇒ (x + 5/6)2 = 43/3 + 25/36
    ⇒ (x + 5/6)2 = 516 + 25/36 = 541/36


    Now this means x + 5/6 is NOT a perfect square. And that means the three consecutive positive integers will also not be positive integers.

    That is my predicament, of which I seek riddance.
     
  2. jcsd
  3. Oct 20, 2012 #2

    tiny-tim

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    welcome to pf!

    hi aaronstonedd! welcome to pf! :smile:
    well, that can't be right …

    the first two terms are one odd and one even, and the third term is even, so the total is odd, ≠ 46 :redface:

    perhaps it means the product of the outer two?

    (and perhaps it means 96 ?)
     
  4. Oct 20, 2012 #3
    Re: Quadratic Equations: Finding three consecutive positive integers given sum of squ

    So you're saying that even + odd + even = odd. That's right, because even + odd = odd. But it's also possible for three consecutive positive integers to be odd + even + odd which is equal to even!

    Is the above argument valid?
     
  5. Oct 20, 2012 #4

    ehild

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    Re: Quadratic Equations: Finding three consecutive positive integers given sum of squ

    Are you sure that the sum is not 45?

    ehild
     
  6. Oct 20, 2012 #5
    Re: Quadratic Equations: Finding three consecutive positive integers given sum of squ

    To ehild: Yes, I'm sure. Only the answer is given, from where I've got the question. The answer given is 4,5,6 although I'm not sure how that fits into the question's conditions.

    The main problem I'm facing is interpreting the question and representing it mathematically.
     
  7. Oct 20, 2012 #6

    tiny-tim

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    1. Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46. Find the numbers.

    4*4 + 5*6 = 46 :wink:
     
  8. Oct 20, 2012 #7

    ehild

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    Re: Quadratic Equations: Finding three consecutive positive integers given sum of squ

    Ingenious! :cool:

    ehild
     
  9. Oct 20, 2012 #8
    Re: Quadratic Equations: Finding three consecutive positive integers given sum of squ

    Thank you tiny-tim, for correcting the question. I've often noticed that if a question seems absurd or solving it gives odd results, then something in the question is probably wrong :wink:.

    My query is solved. Thanks all!
     
    Last edited: Oct 20, 2012
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