Multiple Choice Question (MCQ)

Which of the following integer values of n produce a perfect integer square for n^2+9n+14?

  1. ×

    1

  2. 2

  3. ×

    5

  4. ×

    36

Collected in the board: Perfect square integer

Steven Zheng posted 1 month ago


Answer

  1. Let m\in Z such that

    n^2+9n+14=m^2,

    A perfect square number can be expressed by a smaller perfect number plus an integer, which can be further expressed as the product of two inters using the difference of two squares.


    4n^2+36n+4\times 14=4m^2

    4n^2+36n+81+4\times 14=4m^2+81

    (2n+9)^2-4m^2=81-56 = 25

    (2n+9-2m)(2n+9-2m) = 25

    Let a = 2n+9-2m, b = 2n+9-2m

    Addition of above equations gives

    n = \dfrac{a+b-18}{4}

    and

    a\cdotp b =1\times 25=5\times 5 = 25

    When a = 1, b=25,

    n =\dfrac{26-18}{4} = 2

    n^2+9n+14 = 2^2+9\times 2+14=36

    m = 6

    When

    a = 5, b=5,
    (1)
    is negative, cancel

    So B is the choice

Steven Zheng posted 1 month ago

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