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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 94
Chapter 2, Problem 94

In Exercises 91–100, find all values of x satisfying the given conditions. y=x−x−2andy=4y = x - \(\sqrt{x - 2}\) \(\quad\) \(\text{and}\) \(\quad\) y = 4

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Start with the given system of equations: \(y = x - \sqrt{x - 2}\) and \(y = 4\).
Since both expressions equal \(y\), set them equal to each other: \(4 = x - \sqrt{x - 2}\).
Isolate the square root term: \(\sqrt{x - 2} = x - 4\).
Square both sides to eliminate the square root: \((\sqrt{x - 2})^2 = (x - 4)^2\), which simplifies to \(x - 2 = (x - 4)^2\).
Expand the right side and rearrange the equation to form a quadratic: \(x - 2 = x^2 - 8x + 16\), then bring all terms to one side to get \(0 = x^2 - 9x + 18\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Equations Involving Square Roots

When an equation includes a square root, isolate the root expression and then square both sides to eliminate the root. This process may introduce extraneous solutions, so all potential solutions must be checked in the original equation.
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Solving Quadratic Equations by the Square Root Property

Domain Restrictions for Square Root Functions

The expression inside a square root must be non-negative for real-valued functions. For y = x - √(x - 2), the domain requires x - 2 ≥ 0, so x ≥ 2. This restriction limits the possible values of x when solving the equation.
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Domain Restrictions of Composed Functions

Checking Solutions for Extraneous Roots

After solving equations involving square roots, some solutions may not satisfy the original equation due to the squaring step. Substitute each solution back into the original equation to verify its validity and discard any extraneous roots.
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Imaginary Roots with the Square Root Property
Related Practice
Textbook Question

Solve each equation in Exercises 83–108 by the method of your choice. 3x2−12x+12=03x^2 - 12x + 12 = 0

Textbook Question

In Exercises 95–99, perform the indicated operations and write the result in standard form. 4/(2 + i)(3 - i)

Textbook Question

Solve each equation in Exercises 83–108 by the method of your choice. (3x - 4)2 = 16

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Textbook Question

Solve each absolute value inequality. 4 + |3 - x/3| ≥ 9

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Textbook Question

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. y1=x2+3,y2=x3+52,andy1≤y2y_1 = \(\frac{x}{2}\) + 3, \(\quad\) y_2 = \(\frac{x}{3}\) + \(\frac{5}{2}\), \(\quad\) \(\text{and}\) \(\quad\) y_1 \(\leq\) y_2

Textbook Question

Solve the equations containing absolute value in Exercises 94–95. |2x+1| = 7

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