# How do you solve e^(2x)-(4e^x)+3=0?

Jun 20, 2018

$x = 0 \mathmr{and} x = \ln \left(3\right)$

#### Explanation:

Let $u = {e}^{x}$. Then the equation will become ${u}^{2} - 4 u + 3 = 0$.

This equation can be solved by factoring

$\left(u - 3\right) \left(u - 1\right) = 0$
#u = 3 or 1

Now we see that

${e}^{x} = 3 \mathmr{and} {e}^{x} = 1$
$x = \ln 3 \mathmr{and} x = 0$

Hopefully this helps!

Jun 20, 2018

$x = 0 \mathmr{and} \ln 3 \approx 1.09861$

#### Explanation:

${e}^{2 x} - 4 {e}^{x} + 3 = 0$

Let $\phi = {e}^{x} \to x = \ln \phi$

$\therefore {\phi}^{2} - 4 \phi + 3 = 0$

$\left(\phi - 3\right) \left(\phi - 1\right) = 0$

Hence, $\phi = 1 \mathmr{and} 3$

$\phi = 1 \to x = \ln 1 = 0$

$\phi = 3 \to x = \ln 3 \approx 1.09861$