Textbook Question
If 5 times a number is decreased by 4, the principal square root of this difference is 2 less than the number. Find the number(s).
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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 108
If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).
Verified step by step guidance1
Let the number be represented by the variable \(x\).
Translate the problem statement into an equation: "If a number is decreased by 3" becomes \(x - 3\), and "the principal square root of this difference is 5 less than the number" becomes \(\sqrt{x - 3} = x - 5\).
Square both sides of the equation to eliminate the square root: \((\sqrt{x - 3})^2 = (x - 5)^2\), which simplifies to \(x - 3 = (x - 5)^2\).
Expand the right side: \((x - 5)^2 = x^2 - 10x + 25\), so the equation becomes \(x - 3 = x^2 - 10x + 25\).
Rearrange the equation to standard quadratic form by moving all terms to one side: \(0 = x^2 - 10x + 25 - x + 3\), which simplifies to \(0 = x^2 - 11x + 28\). This quadratic equation can now be solved for \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Setting up Algebraic Equations
This involves translating a word problem into an algebraic equation using variables to represent unknown quantities. Identifying relationships described in the problem allows you to form equations that can be solved systematically.
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05:09
Introduction to Algebraic Expressions
Square Roots and Principal Square Root
The principal square root of a number is the non-negative root. Understanding how to work with square roots, including isolating the root and squaring both sides of an equation, is essential for solving equations involving roots.
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Solving Quadratic Equations
After forming an equation, you may need to rearrange it into a quadratic form and solve using factoring, completing the square, or the quadratic formula. Checking solutions is important because squaring can introduce extraneous roots.
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Related Practice
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 3/(x - 3) + 5/(x - 4) = (x2 - 20)/(x2 - 7x + 12)
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Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 2x/(x - 3) + 6/(x + 3) = - 28/(x2 - 9)
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Textbook Question
In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = x2 - 4x - 5
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Textbook Question
When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.
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Textbook Question
In Exercises 107–110, use graphs to find each set. [1,3) ∩ (0,4)
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