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

Verified step by step guidance

1. Recognize that the equation involves exponential expressions with the same base, \(e\). The equation is \$2e^{2x} + e^x = 6$.
2. Rewrite \(e^{2x}\) as \((e^x)^2\) to transform the equation into a quadratic form in terms of \(e^x\). This gives \$2(e^x)^2 + e^x = 6$.
3. Let \(u = e^x\). Substitute into the equation to get a quadratic equation: \$2u^2 + u = 6$.
4. Rearrange the quadratic equation to standard form: \$2u^2 + u - 6 = 0$.
5. Solve the quadratic equation for \(u\) using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=2\), \(b=1\), and \(c=-6\). After finding the values of \(u\), substitute back \(u = e^x\) and solve for \(x\) by taking the natural logarithm: \(x = \ln(u)\).

Key Concepts

Exponential Functions

Exponential functions involve variables in the exponent, such as e^x, where e is Euler's number (~2.718). Understanding their properties, like growth behavior and how to manipulate expressions with exponents, is essential for solving equations involving terms like e^x and e^{2x}.

Substitution Method for Solving Equations

When an equation contains terms like e^{2x} and e^x, substitution can simplify it by letting a new variable represent e^x. This transforms the equation into a quadratic form, making it easier to solve using algebraic methods before back-substituting to find x.

Solving Quadratic Equations

After substitution, the equation often becomes quadratic, requiring methods like factoring, completing the square, or the quadratic formula to find solutions. Understanding how to solve quadratics is crucial to determine the values of the substituted variable and ultimately solve for x.