Textbook Question
You invested \$30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was \$705.88, how much was invested at each rate?
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 21a
Solve each equation in Exercises 15–34 by the square root property. (x + 2)2 = 25
Verified step by step guidance1
Start by applying the square root property to both sides of the equation. The square root property states that if \((a)^2 = b\), then \(a = \pm \sqrt{b}\). For this problem, take the square root of both sides: \(\sqrt{(x + 2)^2} = \pm \sqrt{25}\).
Simplify both sides of the equation. The square root of \((x + 2)^2\) is \(x + 2\), and the square root of 25 is 5. This gives \(x + 2 = \pm 5\).
Split the equation into two separate cases to account for the \(\pm\) symbol. Case 1: \(x + 2 = 5\). Case 2: \(x + 2 = -5\).
Solve each case for \(x\). For Case 1: Subtract 2 from both sides to get \(x = 5 - 2\). For Case 2: Subtract 2 from both sides to get \(x = -5 - 2\).
Write the solutions as a set. The solutions are the values of \(x\) obtained from both cases.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Root Property
The square root property states that if x^2 = a, then x = ±√a. This property allows us to solve quadratic equations by taking the square root of both sides, leading to two possible solutions: one positive and one negative. It is essential for solving equations where the variable is squared.
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Isolating the Variable
Isolating the variable involves rearranging an equation to get the variable on one side and all other terms on the opposite side. In the context of the square root property, this often means first simplifying the equation to the form x^2 = a before applying the square root. This step is crucial for correctly applying the property.
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Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique is particularly useful when the equation is not in standard form. It helps in deriving the square root property and can also be used to derive the quadratic formula.
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Related Practice
Textbook Question
In Exercises 15–26, use graphs to find each set. (- ∞, 5) ⋃ [1, 8)
Textbook Question
Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 7x + 13 = 2(2x-5) + 3x + 23
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Textbook Question
In Exercises 1–26, solve and check each linear equation. 2 - (7x + 5) = 13 - 3x
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Textbook Question
In Exercises 21–28, divide and express the result in standard form. 2/(3 - i)
Textbook Question
You invested \$20,000 in two accounts paying 1.45% and 1.59% annual interest. If the total interest earned for the year was \$307.50, how much was invested at each rate?