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. 5^2x + 3(5^x) = 28

Verified step by step guidance

1. Recognize that the equation involves exponential expressions with the same base, 5. The equation is \$5^{2x} + 3(5^x) = 28$.
2. Rewrite \$5^{2x}\( as \)(5^x)^2\( to express the equation in terms of a single variable. Let \)y = 5^x\(, so the equation becomes \)y^2 + 3y = 28$.
3. Rewrite the equation as a quadratic equation by moving all terms to one side: \(y^2 + 3y - 28 = 0\).
4. Solve the quadratic equation \(y^2 + 3y - 28 = 0\) using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=3\), and \(c=-28\).
5. After finding the values of \(y\), substitute back \(y = 5^x\) and solve for \(x\) by taking the logarithm base 5: \(x = \log_5(y)\). Calculate the decimal values of \(x\) to the nearest thousandth.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position, such as 5^x. Solving these requires understanding how to manipulate and rewrite expressions to isolate the variable, often by expressing terms with the same base or using substitution.

Substitution Method

When an equation contains terms like 5^(2x) and 5^x, substitution simplifies the problem. For example, letting y = 5^x transforms the equation into a quadratic form, making it easier to solve using algebraic methods.

Rounding Irrational Solutions

Some solutions to exponential equations are irrational numbers. These should be approximated as decimals rounded to a specified place value, such as the nearest thousandth, to provide a practical and understandable answer.