Solve each equation in Exercises 41–60 by making an appropriate substitution. x^(-2) - x^(-1) - 6 = 0

Verified step by step guidance

1. Recognize that the equation involves negative exponents. Rewrite the terms using positive exponents: \( x^{-2} = \frac{1}{x^2} \) and \( x^{-1} = \frac{1}{x} \). The equation becomes \( \frac{1}{x^2} - \frac{1}{x} - 6 = 0 \).
2. To simplify the equation, make a substitution. Let \( u = \frac{1}{x} \). This means \( \frac{1}{x^2} = u^2 \). Substituting these into the equation gives \( u^2 - u - 6 = 0 \).
3. Now solve the quadratic equation \( u^2 - u - 6 = 0 \). Factorize the quadratic equation, if possible, or use the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = -6 \).
4. Once you find the values of \( u \), substitute back \( u = \frac{1}{x} \) to return to the original variable \( x \). Solve for \( x \) by taking the reciprocal of each \( u \) value: \( x = \frac{1}{u} \).
5. Check all solutions in the original equation \( x^{-2} - x^{-1} - 6 = 0 \) to ensure they are valid and do not result in division by zero. Discard any extraneous solutions if necessary.

Key Concepts

Exponential Functions and Negative Exponents

Exponential functions involve expressions where a variable is raised to a power. Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, x^(-1) is equivalent to 1/x, and x^(-2) is equivalent to 1/x^2. Understanding this concept is crucial for simplifying equations that contain negative exponents.

Substitution Method

The substitution method is a technique used to simplify complex equations by replacing a variable or expression with a single variable. In this case, substituting x^(-1) with a new variable, such as y, can transform the equation into a quadratic form, making it easier to solve. This method is particularly useful for equations involving powers and can streamline the solving process.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, or the quadratic formula. Recognizing the transformed equation as a quadratic is essential for applying these methods effectively to find the values of the original variable.