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If the roots of x^2 - px + q = 0 are two consecutive integers, prove that p^2 - 4q -1 = 0

  • maths -

    sum = p
    product = q
    let a & b the roots
    a + b = p
    ab = q
    since a & b are consecutive integers (i.e. b = a+1):
    2a + 1 = p
    a^2 + a = q
    substitute.
    p^2 - 4q -1 = 0
    4a^2 + 4a + 1 - 4a^2 + 4a - 1 = 0
    indeed,
    0 = 0

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