Math

Find the value of K such that the following trinomials can be factored over the integers:

1. 36x^2+18x+K

2. 3x^2 - 16x+K

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  1. let's look at the discriminant.
    if b^2 - 4ac is a perfect square, then it can be factored over the rationals, so we start with that.

    I will do the 2nd, since it has smaller numbers

    for 3x^2 - 16x + k
    b^2 - 4ac
    = 256-12k
    = 4(64-3k)
    remember we have to take the square root of that
    √(4(64-3k))
    = 2√(64-3k)
    For 64-3k to be a perfect square
    we need 3k to be 0,15,28,39,48,55,60 or 63
    of those only 0,15,39,48,60, and 63 are multiples of 3
    So for rationals, k could be 0, 5, 13, 16, 20 or 21
    testing:
    let's try k = 13
    3x^2 - 16x + 13
    x = (-16 ± √100)/6
    = -1 or 13/3

    so 3x^2 - 16x + 13 = (x+1)(3x-13)

    k = 0, 5, 13, 16, 20, or 21

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