Textbook Question
Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x5+2x4−15x3−10x2+12x+8
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Ch. 3 - Polynomial and Rational Functions
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 4, Problem 59
Solve:
Verified step by step guidance1
Start with the given equation: \(\sqrt{x} + \sqrt{x + 5} = 5\).
Isolate one of the square root terms, for example, \(\sqrt{x + 5} = 5 - \sqrt{x}\).
Square both sides of the equation to eliminate the square root on the right: \(\left(\sqrt{x + 5}\right)^2 = \left(5 - \sqrt{x}\right)^2\).
Simplify both sides: \(x + 5 = 25 - 10\sqrt{x} + x\).
Rearrange the equation to isolate the remaining square root term and then square both sides again to solve for \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Roots and Radicals
Square roots represent a value that, when multiplied by itself, gives the original number. Understanding how to manipulate and simplify expressions involving square roots is essential, especially when they appear in equations. Recognizing that √x denotes the principal (non-negative) root is important for solving radical equations.
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Imaginary Roots with the Square Root Property
Isolating and Squaring Both Sides
To solve equations involving square roots, one common method is to isolate the radical expression and then square both sides to eliminate the square root. This process can introduce extraneous solutions, so it is crucial to check all potential solutions in the original equation.
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Solving Quadratic Equations
After squaring, the equation often becomes quadratic in form. Knowing how to solve quadratic equations using factoring, completing the square, or the quadratic formula is necessary to find the values of x. Verifying solutions against the original equation ensures only valid answers are accepted.
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06:08Solving Quadratic Equations by Factoring
Related Practice
Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x - 2)/(x + 2) ≤ 2
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Textbook Question
In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (2x - 4)/(x + 3)
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Textbook Question
In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)
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Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation.1/(x - 3) < 1
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Textbook Question
Follow the seven steps to graph each rational function. f(x)=2x/(x2−4)
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