Solve Exponential 2B2F68

1. **State the problem:** Solve the equation $$e^{2x} - 6e^x + 5 = 0$$ for $x$. 2. **Rewrite the equation:** Let $u = e^x$. Then the equation becomes $$u^2 - 6u + 5 = 0$$. 3. **Solve the quadratic equation:** Use the quadratic formula $$u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-6$, and $c=5$. 4. **Calculate the discriminant:** $$\sqrt{(-6)^2 - 4 \times 1 \times 5} = \sqrt{36 - 20} = \sqrt{16} = 4$$. 5. **Find the roots:** $$u = \frac{6 \pm 4}{2}$$ 6. **Calculate each root:** - $$u_1 = \frac{6 + 4}{2} = \frac{10}{2} = 5$$ - $$u_2 = \frac{6 - 4}{2} = \frac{2}{2} = 1$$ 7. **Back-substitute for $x$:** Recall $u = e^x$, so - For $u_1 = 5$: $$e^x = 5 \implies x = \ln(5)$$ - For $u_2 = 1$: $$e^x = 1 \implies x = \ln(1) = 0$$ **Final answer:** $$x = \ln(5) \text{ or } x = 0$$