Consider the two equations below. Explain completely the similarities and differences in how you would solve each e

Question

Consider the two equations below. Explain completely the similarities and

differences in how you would solve each equation. Be clear and complete.

3* = 12 and x3 = 12

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Genesis 2 months 2021-09-17T16:59:01+00:00 1 Answer 0 views 0

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    2021-09-17T17:00:04+00:00

    Answer:

    Here, the given equations,

    3^x=12 and x^3=12

    Similarity: In both equations there is only one variable ( i.e. x)

    Difference: 3^x=12 is an exponential equation while x^3=12 is a polynomial equation.

    Now, when we solve an exponential equation we take log in both sides of the equation as follows:

    3^x=12

    \log(3^x)=\log12

    x\log 3 =\log 12       ( ∵ \log m^n=n\log m )

    \implies x =\frac{\log 12}{\log 3}

    Hence, the solution of the equation 3^x=12 is x=\frac{\log 12}{\log 3}.

    While, when we solve a polynomial we find the roots as follows:

    x^3=12

    x^3-12=0

    x^3-(12^\frac{1}{3})^3=0

    (x-12^\frac{1}{3})(x^2+12^\frac{1}{3}x+(12^\frac{1}{3})^2)=0

    By zero product property,

    (x-12^\frac{1}{3})=0 or (x^2+12^\frac{1}{3}x+(12^\frac{1}{3})^2)=0

    If (x-12^\frac{1}{3})=0, then x=12^\frac{1}{3}

    If x^2+12^\frac{1}{3}x+(12^\frac{1}{3})^2=0,

    Then, by quadratic formula,

    x=\frac{-12^\frac{1}{3}\pm \sqrt{12^\frac{2}{3}-4(1)(12^\frac{1}{3})^2}}{2}

     =\frac{-12^\frac{1}{3}\pm \sqrt{12^\frac{2}{3}-4(12^\frac{2}{3})}}{2}

     =\frac{-12^\frac{1}{3}\pm i\sqrt{3(12^\frac{2}{3})}}{2}

     =12^\frac{1}{3}(\frac{-1\pm i\sqrt{3}}{2})

    Hence, the solutions of the equation x^3=12 are 12^\frac{1}{3},  12^\frac{1}{3}(\frac{-1+i\sqrt{3}}{2})  and  12^\frac{1}{3}(\frac{-1-i\sqrt{3}}{2}) .

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