MATHS URGENT

By making the substitution u =ln x show that the following equation has exactly one real-number solution:

9/ln x + ln x^4 =-12

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  1. 9/u + 4u = -12

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  2. ln ( x ^ 4) = 4 ln x

    9 / ln x + ln ( x ^ 4 ) = - 12

    9 / u + 4 u = - 12 Multiply both sides by u

    9 u / u + 4 u * u = ( - 12 ) * u

    9 + 4 u ^ 2 = - 12 u Add 12 u to both sides

    9 + 4 u ^ 2 + 12 u = - 12 u + 12 u

    4 u ^ 2 + 12 u + 9 = 0

    Now, find discriminant using formula D = b ^ 2 − 4 a c

    In this case: a = 4, b = 12 , c = 9

    D = 12 ^ 2 - 4 * 4 * 9 = 144 - 144 = 0

    Since discriminant is zero, then there is one root.

    To find it, use formula u = - b / 2 a = - 12 / 2 * 4 = - 12 / 8 = - 4 * 3 / 4 * 2 = - 3 / 2

    u = ln x

    so

    ln x = - 3 / 2

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