Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Choosing a Method to Solve Quadratics

College Algebra

1. Equations & Inequalities

Choosing a Method to Solve Quadratics

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1:11 minutes

Problem 9

Textbook Question

Match each equation in Column I with the correct first step for solving it in Column II. (x+5)^2/3 - (x+5)^1/3 - 6 = 0

Verified step by step guidance

Identify the substitution: Let \( y = (x+5)^{1/3} \).

Rewrite the equation using the substitution: \( y^2 - y - 6 = 0 \).

Recognize that the equation is now a quadratic in terms of \( y \).

Factor the quadratic equation: \( (y - 3)(y + 2) = 0 \).

Solve for \( y \) by setting each factor equal to zero: \( y - 3 = 0 \) and \( y + 2 = 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponents and Roots

Understanding exponents and roots is crucial for manipulating equations involving powers. In the given equation, (x+5) is raised to the power of 2/3 and 1/3, which indicates that we are dealing with both squaring and taking cube roots. Recognizing how to simplify these expressions will help in isolating the variable effectively.

Substitution

Substitution is a technique used to simplify complex expressions by replacing a variable or expression with a single variable. In this case, letting u = (x + 5)^(1/3) can transform the equation into a more manageable form, allowing for easier solving of the polynomial equation that results.

Factoring Polynomials

Factoring polynomials is a method used to break down complex polynomial expressions into simpler components that can be solved more easily. Once the equation is simplified through substitution, recognizing patterns or applying the quadratic formula can help in finding the roots of the resulting polynomial equation.