Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4.e^2x - 6e^x + 8 = 0

Verified step by step guidance

1. Let \( u = e^x \). This substitution transforms the equation into a quadratic form: \( u^2 - 6u + 8 = 0 \).
2. Factor the quadratic equation: \( (u - 2)(u - 4) = 0 \).
3. Set each factor equal to zero and solve for \( u \): \( u - 2 = 0 \) or \( u - 4 = 0 \).
4. Solve for \( u \): \( u = 2 \) or \( u = 4 \).
5. Substitute back \( u = e^x \) and solve for \( x \): \( e^x = 2 \) or \( e^x = 4 \). Use natural logarithms to find \( x \).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as e^x. To solve these equations, one often uses substitution to simplify the expression. For example, if we let y = e^x, the equation can be transformed into a quadratic form, making it easier to solve.

Quadratic Formula

The quadratic formula is a method for solving quadratic equations of the form ax^2 + bx + c = 0. It states that the solutions for x can be found using x = (-b ± √(b² - 4ac)) / (2a). This formula is essential for finding exact solutions, especially when the equation does not factor easily.

Irrational and Exact Solutions

Irrational solutions are those that cannot be expressed as simple fractions and often involve square roots or other non-repeating decimals. In contrast, exact solutions are expressed in their simplest radical form. Understanding how to convert between these forms is crucial for providing answers in the required format.