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Math Help - complicated equation

  1. #1
    Junior Member
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    complicated equation

    need help with this one :/

    2^(3-x) = 2^x - 2

    2^(3-x) - 2^x + 2 = 0 = 2^-x * 2 * 2 * 2 - 2^x + 2

    someone who can help with the rest ?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    2^{3-x}=2^x-2\;\Leftrightarrow\;\dfrac{2^3}{2^x}=2^x-2

    Now, use the substitution u=2^x

    Fernando Revilla
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  3. #3
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    Hello, paulaa!

    Another approach . . .


    2^{3-x} \:=\: 2^x - 2

    Multiply through by 2^x:\;\;2^3 \:=\:2^{2x} - 2\!\cdot\!2^x \quad\Rightarrow\quad 2^{2x} - 2\!\cdot\!2^x - 8 \:=\:0


    Factor: . (2^x + 2)(2^x-4) \:=\:0


    and we have: . \begin{Bmatrix}2^x+2 \:=\:0 & \Rightarrow & 2^x \,=\,\text{-}2 & \Rightarrow & \text{no real roots} \\ \\[-3mm]<br />
2^x - 4 \:=\:0 & \Rightarrow & 2^x \,=\,4 & \Rightarrow & \boxed{x \,=\,2} \end{Bmatrix}

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  4. #4
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    It's worth noting that Fernando and Soroban's methods are essentially the same. Soroban just didn't formally make a substutution. Most students seem to understand the formal substitution better at first, and then switch to what Soroban did after they've done enough problems.
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