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March 27, 2017

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How many ordered pairs of integers (x,y) are there such that 2x2−2xy+y2=225?

  • math - ,

    using the quadratic equation, I solved for y
    where a = 1 , b = -2x and c = 2x^2-225
    y = (2x ± √(4x^2 - 4(4x^2 - 225) )/2
    = (2x ± √(900 - 20x^2)/2
    = x ± √(225 - x^2)

    now we want x and y to be integers, so
    225 - x^2 must be a perfect square

    so possible results for 225- a perfect square giving us a perfect square are
    225 - 0 = 225 , good one
    225 - 1
    225 - 4
    225 - 9
    225 - 16
    225 - 25
    225 - 36
    225 - 49
    225 - 64
    225 - 81 = 144 ahhhh
    225 - 100
    225 -121
    225 - 144 = 81 --- another one
    225 - 169
    225 - 196
    225- 225 = 0 --- that one works

    so we have x = 0, ±9, ±12, ±15
    each one will give an integer for y
    (0, ±15) , (9, 21), (9, -3), (-9, 3), (-9, -21) ....

    so I count 14 such ordered pairs

  • math - ,

    brilliant says its wrong... 14 is wrong... it meant 2x^2−2xy+y^2=225

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