Textbook Question
In Exercises 61–66, find all values of x satisfying the given conditions. y1 = 5/(x + 4), y2 = 3/(x + 3), y3 = (12x + 19)/(x2 + 7x + 12). and y1 + y2 = y3.
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 65
Solve each equation in Exercises 65–74 using the quadratic formula.
Verified step by step guidance1
Identify the coefficients in the quadratic equation \(x^2 + 8x + 15 = 0\). Here, \(a = 1\), \(b = 8\), and \(c = 15\).
Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}\).
Simplify inside the square root (the discriminant): calculate \(8^2 - 4 \cdot 1 \cdot 15\) to determine the value under the square root.
Evaluate the square root and then compute the two possible values for \(x\) by using the plus and minus signs in the formula.
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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 second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed, and its solutions are the x-values where the parabola intersects the x-axis.
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Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses coefficients a, b, and c to find roots, including real and complex solutions.
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Discriminant
The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.
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Related Practice
Textbook Question
Perform the indicated operation(s) and write the result in standard form. (2 - 3i)(1 - i) - (3 - i)(3 + i)
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Textbook Question
In Exercises 59–94, solve each absolute value inequality. |2(x - 1) + 4| ≤ 8
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Textbook Question
Solve each equation in Exercises 47–64 by completing the square. 3x2 - 5x - 10 = 0
Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| = 5
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Textbook Question
Perform the indicated operation(s) and write the result in standard form. (8 + 9i)(2 - i) - (1 - i)(1 + i)