Textbook Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the ordered pairs (- 2, 2), (0, 0), and (2, 2) to graph a straight line.
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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 69
Solve each equation in Exercises 65–74 using the quadratic formula.
Verified step by step guidance1
Identify the coefficients in the quadratic equation \(3x^2 - 3x - 4 = 0\). Here, \(a = 3\), \(b = -3\), and \(c = -4\).
Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Calculate the discriminant \(\Delta = b^2 - 4ac\) by substituting the values: \(\Delta = (-3)^2 - 4(3)(-4)\).
Substitute \(a\), \(b\), and the discriminant \(\Delta\) into the quadratic formula: \(x = \frac{-(-3) \pm \sqrt{\Delta}}{2(3)}\).
Simplify the expression under the square root and the entire fraction to express the two possible solutions for \(x\).
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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, b, and c are constants and a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.
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Introduction to Quadratic Equations
Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots, including complex solutions when the discriminant is negative.
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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
In Exercises 67–70, find all values of x such that y = 0.
y = 2[3x - (4x - 6)] - 5(x - 6)
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Textbook Question
In Exercises 67–70, find all values of x such that y = 0.
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Textbook Question
In Exercises 59–94, solve each absolute value inequality. |x| > 3
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Textbook Question
In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. 5√-16 + 3√-81
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Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 7|5x| + 2 = 16
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