How can I solve for $x$ $$3e^{-2x}-6e^{-x}=72$$
If someone can go step by step and do the problem, it would be really appreciable.
Solve for $x$ without using a calculator.
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$\begingroup$
calculus
exponential-function
2 Answers
7
Substitute $e^{-x}=t$.
Now, Note that since $e^{-2x}=\left( e^{-x} \right)^2$, we have that $$3e^{-2x}-6e^{-x}-72=0 \iff 3t^2-6t-72=0 \iff 3(t+4)(t-6)=0$$ From factorization. Since $t>0$, we have that $e^{-x}=t=6$.
So $-x=\ln 6$, or $x=-\ln 6$.
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0Guess that's the expected answer, but does it really count for `without using a calculator` ;-) – 2017-02-14
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0@dxiv Yes, why don't you think it doesn't? It doesn't ask to find the decimal expansion of $-\ln 6$. I think I'm missing something. – 2017-02-14
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1I wasn't totally serious (+1), but then if I otherwise proved that the equation has one unique solution, I wouldn't expect *that* to count for an answer, though without a calculator it's just as fuzzily defined as $\ln 6\,$. – 2017-02-14
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1@dxiv OK, I'll think on that. – 2017-02-14
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Hint: substitute $e^{-x} = t$ to obtain a quadratic equation in t.