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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1:44 minutes

Problem 95

Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties ofabsolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 canbe solved by inspection.) | 4x^2 - 23x - 6 | = 0

Verified step by step guidance

Recognize that the equation involves an absolute value: \(|4x^2 - 23x - 6| = 0\).

Understand that the absolute value of an expression is zero only when the expression itself is zero.

Set the expression inside the absolute value equal to zero: \(4x^2 - 23x - 6 = 0\).

This is a quadratic equation, so apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -23\), and \(c = -6\).

Calculate the discriminant \(b^2 - 4ac\) and then use it to find the values of \(x\) using the quadratic formula.

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Key Concepts

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

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. In equations, this means that |A| = B implies A = B or A = -B, which is crucial for solving equations involving absolute values.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding how to manipulate and solve these equations is essential for addressing the given problem.

Inspection Method

The inspection method involves solving equations or inequalities by identifying solutions through observation rather than formal algebraic manipulation. This approach is particularly useful for simpler equations or when the solutions are evident. In the context of the given problem, recognizing that the absolute value equation can yield straightforward solutions by considering the nature of the quadratic expression is key.