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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 59
Capitolo 4, Problema 59

Solve: x+x+5=5\(\sqrt{x}\) + \(\sqrt{x + 5}\) = 5

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Start with the given equation: \(\sqrt{x} + \sqrt{x + 5} = 5\).
Isolate one of the square root terms, for example, \(\sqrt{x + 5} = 5 - \sqrt{x}\).
Square both sides of the equation to eliminate the square root on the right: \(\left(\sqrt{x + 5}\right)^2 = \left(5 - \sqrt{x}\right)^2\).
Simplify both sides: \(x + 5 = 25 - 10\sqrt{x} + x\).
Rearrange the equation to isolate the remaining square root term and then square both sides again to solve for \(x\).

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Square Roots and Radicals

Square roots represent a value that, when multiplied by itself, gives the original number. Understanding how to manipulate and simplify expressions involving square roots is essential, especially when they appear in equations. Recognizing that √x denotes the principal (non-negative) root is important for solving radical equations.
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Isolating and Squaring Both Sides

To solve equations involving square roots, one common method is to isolate the radical expression and then square both sides to eliminate the square root. This process can introduce extraneous solutions, so it is crucial to check all potential solutions in the original equation.
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Solving Quadratic Equations

After squaring, the equation often becomes quadratic in form. Knowing how to solve quadratic equations using factoring, completing the square, or the quadratic formula is necessary to find the values of x. Verifying solutions against the original equation ensures only valid answers are accepted.
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