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11 December, 07:56

Prove that the expression 2x (x-6) - 3 (x2-4x+1) is negative for all values of x.

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  1. 11 December, 10:17
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    Simplify the following:

    2 x (x - 6) - 3 (x^2 - 4 x + 1)

    2 x (x - 6) = 2 x^2 - 12 x:

    2 x^2 - 12 x - 3 (x^2 - 4 x + 1)

    -3 (x^2 - 4 x + 1) = - 3 x^2 + 12 x - 3:

    -3 x^2 + 12 x - 3 + 2 x^2 - 12 x

    Grouping like terms, 2 x^2 - 3 x^2 + 12 x - 12 x - 3 = (2 x^2 - 3 x^2) - 3 + (-12 x + 12 x):

    (2 x^2 - 3 x^2) - 3 + (-12 x + 12 x)

    2 x^2 - 3 x^2 = - x^2:

    -x^2 - 3 + (-12 x + 12 x)

    12 x - 12 x = 0:

    -x^2 - 3

    Factor - 1 out of - x^2 - 3:

    Answer: - (x^2 + 3)
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