Domanda del libro di testo
Solve each equation in Exercises 83–108 by the method of your choice. 2x2 - x = 1
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 2, Problema 83
The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84.
Guida verificata passo dopo passo1
Recognize that the equation |2x^2 - 4| = |2x^2| fits the form |u| = |v|, where u = 2x^2 - 4 and v = 2x^2.
Apply the property that |u| = |v| implies u = v or u = -v. This gives two separate equations to solve:
\[2x^2 - 4 = 2x^2\] and \[2x^2 - 4 = -2x^2\].
Solve the first equation \[2x^2 - 4 = 2x^2\] by isolating terms and simplifying to find possible values of x.
Solve the second equation \[2x^2 - 4 = -2x^2\] by moving all terms to one side to form a quadratic equation, then solve for x.
Check all solutions in the original equation to ensure they satisfy the absolute value equality.
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Absolute Value Definition and Properties
Absolute value represents the distance of a number from zero on the number line, always non-negative. For any expression u, |u| = u if u ≥ 0, and |u| = -u if u < 0. Understanding this helps rewrite equations involving absolute values into equivalent forms without the bars.
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Change of Base Property
Equations Involving Two Absolute Values
When an equation has two absolute values set equal, such as |u| = |v|, it can be rewritten as two separate equations: u = v or u = -v. This property allows solving complex absolute value equations by breaking them into simpler cases.
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05:28Equations with Two Variables
Solving Quadratic Equations
The expressions inside the absolute values may be quadratic, requiring techniques like factoring, using the quadratic formula, or simplifying to find solutions. Recognizing and solving these quadratic equations is essential to find all possible values of the variable.
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06:08Solving Quadratic Equations by Factoring
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