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the sum of a number and its square is one and a half times their difference. what is the number?

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  1. direct translation of the English to Math:

    x + x^2 = 1.5|x^2 - x|

    case1:
    times 2
    2x + 2x^2 = 3x^2 - 3x
    x^2 - 5x = 0
    x(x-5) = 0
    x = 0 or x = 5

    or

    case2:
    x+x^2 = 1.5(x-x^2)
    times 2
    2x + 2x^2 = 3x - 3x^2
    5x^2 - x = 0
    x(5x-1) = 0
    x = 0 or x = 1/5

    so x = 0 , (the trivial case)
    or x = 5 or x = 1/5

    check:
    the x = 0 case we can check mentally
    if x = 5
    sum of squares = 5 + 25 = 30
    their difference (25 - 5) = 20
    1.5 of that = 30 , check!

    if x = 1/5
    sum of squares = 1/5 + 1/25 = 6/25
    1.5(1/5 - 1/25) = 6/25 , check!

    x = 0, 5, 1/5

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