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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 45
Capitolo 2, Problema 45

Solve each equation in Exercises 41–60 by making an appropriate substitution. x - 13√x + 40 = 0

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Identify the substitution to simplify the equation. Since the equation contains both \( x \) and \( \sqrt{x} \), let \( t = \sqrt{x} \). This means \( x = t^2 \).
Rewrite the original equation \( x - 13\sqrt{x} + 40 = 0 \) in terms of \( t \). Substitute \( x = t^2 \) and \( \sqrt{x} = t \) to get \( t^2 - 13t + 40 = 0 \).
Recognize that the equation \( t^2 - 13t + 40 = 0 \) is a quadratic equation in standard form. Use factoring, completing the square, or the quadratic formula to solve for \( t \).
After finding the values of \( t \), recall that \( t = \sqrt{x} \). To find \( x \), square each solution for \( t \) to get \( x = t^2 \).
Check each solution for \( x \) in the original equation to ensure it is valid, especially since \( \sqrt{x} \) implies \( x \geq 0 \). Discard any extraneous solutions.

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Substitution Method

The substitution method involves replacing a complex expression with a simpler variable to transform the equation into a more manageable form. In this problem, letting √x = t simplifies the equation, making it easier to solve for t before back-substituting to find x.
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Choosing a Method to Solve Quadratics

Quadratic Equations

A quadratic equation is a second-degree polynomial equation typically in the form ax² + bx + c = 0. After substitution, the given equation becomes quadratic in terms of the new variable, allowing the use of factoring, completing the square, or the quadratic formula to find solutions.
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Introduction to Quadratic Equations

Domain Restrictions for Square Roots

Since √x represents the principal (non-negative) square root, the variable x must be non-negative (x ≥ 0). Additionally, solutions obtained after substitution must be checked to ensure they satisfy this domain restriction to avoid extraneous roots.
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Imaginary Roots with the Square Root Property
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