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. 2e^2x + e^x = 6

Verified step by step guidance

1. Step 1: Let $u=e^x$. This substitution simplifies the equation to $2u^2+u=6$.
2. Step 2: Rearrange the equation to form a standard quadratic equation: $2u^2+u-6=0$.
3. Step 3: Use the quadratic formula $u=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ where $a=2$, $b=1$, and $c=-6$ to solve for $u$.
4. Step 4: Calculate the discriminant $b^2-4ac$ to determine the nature of the roots.
5. Step 5: Once you find the values of $u$, substitute back $u=e^x$ to solve for $x$ by taking the natural logarithm: $x=\ln (u)$.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay and are characterized by their unique properties, such as the constant ratio of change. Understanding how to manipulate and solve equations involving exponential functions is crucial for finding solutions in the given problem.

Substitution Method

The substitution method is a technique used to simplify complex equations by replacing a variable or expression with a single variable. In the context of the given equation, substituting e^x with a new variable (e.g., y) can transform the equation into a quadratic form, making it easier to solve. This method is particularly useful when dealing with exponential equations, as it reduces the complexity of the problem.

Irrational Numbers and Decimal Approximation

Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning their decimal representation is non-repeating and non-terminating. When solving equations that yield irrational solutions, it is often necessary to provide decimal approximations to a specified degree of accuracy, such as the nearest thousandth. This practice helps in understanding the magnitude of the solutions and is essential for presenting results in a clear and standardized format.