Question

Solve: e^2x=3(e^x)+4

Equations and inequalities
ANSWERED
asked 2021-08-15
Solve:
\(\displaystyle{e}^{{2}}{x}={3}{\left({e}^{{x}}\right)}+{4}\)

Expert Answers (1)

2021-08-16
We have to solve \(\displaystyle{e}^{{2}}{x}={3}{\left({e}^{{x}}\right)}+{4}\) Let \(\displaystyle{u}={e}^{{x}}\)
Then given equation come:
\(\displaystyle{u}^{{2}}={3}{u}+{4}\Rightarrow{u}^{{2}}-{3}{u}-{4}={0}\)
By using quadratic formula we get
\(\displaystyle{u}={\frac{{-{\left(-{3}\right)}±\sqrt{{-{3}}}^{{2}}-{4}\cdot{1}\cdot{\left(-{4}\right)}}}{{{2}\cdot{1}}}}={\frac{{{3}±\sqrt{{{9}+{16}}}}}{{{2}}}}={4},{1}\)
Therefore we get
\(\displaystyle{e}^{{x}}={4}{\quad\text{and}\quad}{e}^{{x}}={1}\)
By taking logarithm in the above two equations we get
\(\displaystyle{x}={\ln{{\left({4}\right)}}}\) and \(\displaystyle{x}={\ln{{\left({1}\right)}}}={0}\)
Notice that x=0 does not satisfy the given equation.
Hence the only solution of the given equation is x=ln(4)=2ln(2)
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