Textbook Question
Solve each equation in Exercises 15–34 by the square root property. 3x2 - 1 = 47
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 19
Solve each equation in Exercises 15–34 by the square root property.
Verified step by step guidance1
Start by isolating the term with the variable squared. Add 5 to both sides of the equation to get: \(2x^2 - 5 + 5 = -55 + 5\), which simplifies to \(2x^2 = -50\).
Next, divide both sides of the equation by 2 to solve for \(x^2\): \(\frac{2x^2}{2} = \frac{-50}{2}\), resulting in \(x^2 = -25\).
Apply the square root property, which states that if \(x^2 = k\), then \(x = \pm \sqrt{k}\). So, take the square root of both sides: \(x = \pm \sqrt{-25}\).
Recognize that \(\sqrt{-25}\) involves the square root of a negative number, which introduces imaginary numbers. Rewrite \(\sqrt{-25}\) as \(\sqrt{25} \times \sqrt{-1}\).
Simplify \(\sqrt{25}\) to 5 and recall that \(\sqrt{-1} = i\), the imaginary unit. Therefore, express the solutions as \(x = \pm 5i\).
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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 x² = k, then x = ±√k. This method is used to solve quadratic equations that can be written in the form x² = a constant. It simplifies solving by isolating the squared term and then taking the square root of both sides.
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Imaginary Roots with the Square Root Property
Isolating the Quadratic Term
Before applying the square root property, the quadratic term must be isolated on one side of the equation. This involves using algebraic operations such as addition, subtraction, multiplication, or division to rewrite the equation in the form x² = number.
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06:36Solving Quadratic Equations Using The Quadratic Formula
Handling Negative Values Under the Square Root
If the value under the square root (the radicand) is negative, the solutions involve imaginary numbers. Understanding complex numbers and the imaginary unit i (where i² = -1) is essential to express solutions when the radicand is less than zero.
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02:20Imaginary Roots with the Square Root Property
Related Practice
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Exercises 19–20 involve markup, the amount added to the dealer's cost of an item to arrive at the selling price of that item. The selling price of a refrigerator is \$1198. If the markup is 25% of the dealer's cost, what is the dealer's cost of the refrigerator?
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Including a 17.4% hotel tax, your room in Chicago cost \$287.63 per night. Find the nightly cost before the tax was added.