Skip to main content
Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblema 43
Capitolo 2, Problema 43

Solve each equation in Exercises 41–60 by making an appropriate substitution. 9x4=25x2−169x^4 = 25x^2 - 16

Guida verificata passo dopo passo
1
Identify the substitution to simplify the equation. Notice that the equation involves terms with \(x^4\) and \(x^2\). Let \(u = x^2\), so that \(x^4 = (x^2)^2 = u^2\).
Rewrite the original equation \(9x^4 = 25x^2 - 16\) in terms of \(u\): it becomes \(9u^2 = 25u - 16\).
Bring all terms to one side to set the equation equal to zero: \(9u^2 - 25u + 16 = 0\).
Solve the quadratic equation \(9u^2 - 25u + 16 = 0\) using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=9\), \(b=-25\), and \(c=16\).
After finding the values of \(u\), substitute back \(u = x^2\) and solve each resulting equation \(x^2 = u\) for \(x\) by taking the square root, remembering to consider both positive and negative roots.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
9m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Polynomial Equations

Polynomial equations involve expressions with variables raised to whole-number exponents. Understanding how to manipulate and solve these equations, especially higher-degree polynomials like quartic (degree 4), is essential for finding their roots or solutions.
Video consigliato:
05:13
Introduction to Polynomials

Substitution Method

The substitution method simplifies complex equations by replacing a part of the equation with a new variable. For example, substituting y = x² transforms a quartic equation into a quadratic one, making it easier to solve.
Video consigliato:
04:03
Choosing a Method to Solve Quadratics

Solving Quadratic Equations

Once substitution reduces the equation to quadratic form, solving it involves techniques like factoring, completing the square, or using the quadratic formula. These methods help find the values of the substituted variable, which can then be back-substituted to find the original variable.
Video consigliato:
06:08
Solving Quadratic Equations by Factoring
Pratica correlata
Domanda del libro di testo

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 1 - x/2 > 4

770
views
Domanda del libro di testo

Use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.

1371
views
Domanda del libro di testo
Solve and check: 24 + 3 (x + 2) = 5(x − 12).
873
views
Domanda del libro di testo

In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 3 - √-7)2

830
views
Domanda del libro di testo

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 7/2x - 5/3x = 22/3

844
views
Domanda del libro di testo

In Exercises 36–43, use the five-step strategy for solving word problems. The length of a rectangular field is 6 yards less than triple the width. If the perimeter of the field is 340 yards, what are its dimensions?

744
views