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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblema 94
Capitolo 2, Problema 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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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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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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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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