Textbook Question
In Exercises 59–94, solve each absolute value inequality. |3x - 8| > 7
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 73a
Solve each equation in Exercises 65–74 using the quadratic formula. x2 - 6x + 10 = 0
Verified step by step guidance1
Identify the coefficients of the quadratic equation in standard form ax^2 + bx + c = 0. Here, a = 1, b = -6, and c = 10.
Recall the quadratic formula: . Substitute the values of a, b, and c into the formula.
Simplify the discriminant, which is the part under the square root: . Compute (b squared) and subtract .
Determine whether the discriminant is positive, zero, or negative. If it is negative, the solutions will involve imaginary numbers. If it is zero, there is one real solution. If it is positive, there are two distinct real solutions.
Simplify the entire expression for x by calculating the numerator and dividing by the denominator. If the discriminant is negative, express the solutions in terms of complex numbers using , where .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equation
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 this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula.
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Quadratic Formula
The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients from the quadratic equation. This formula provides the roots of the equation, which can be real or complex depending on the value of the discriminant (b² - 4ac).
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Discriminant
The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots of the quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. Understanding the discriminant helps in predicting the type of solutions without solving the equation.
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Related Practice
Textbook Question
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair (2, 5) satisfies 3y - 2x = - 4.
Textbook Question
Evaluate (x2 + 19)/(2 - x) for x = 3i.
Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x + 1| + 5 = 3
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Textbook Question
In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4(x + 5) = 21 + 4x
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Textbook Question
Exercises 73–75 will help you prepare for the material covered in the next section. Simplify: √18 - √8
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