Solve each equation by the method of your choice. 1/(x2 - 3x + 2) = 1/(x + 2) + 5/(x2 - 4)

Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 127

2장, 문제 127

Solve each equation by the method of your choice. 1/(x² - 3x + 2) = 1/(x + 2) + 5/(x² - 4)

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Rewrite the equation to have a common denominator on both sides. Start by factoring the denominators where possible. For example, factor \(x^2 - 3x + 2\) as \((x - 1)(x - 2)\) and \(x^2 - 4\) as \((x - 2)(x + 2)\).

Express all terms with the least common denominator (LCD), which is \((x - 1)(x - 2)(x + 2)\). Rewrite each fraction accordingly: \(\frac{1}{(x - 1)(x - 2)}\), \(\frac{1}{x + 2}\), and \(\frac{5}{(x - 2)(x + 2)}\).

Multiply through by the LCD \((x - 1)(x - 2)(x + 2)\) to eliminate the denominators. This will leave you with a polynomial equation to solve.

Simplify the resulting polynomial equation by combining like terms and setting it equal to zero. This will give you a quadratic equation.

Solve the quadratic equation using factoring, the quadratic formula, or completing the square. Be sure to check for any extraneous solutions by substituting back into the original equation, as some solutions may make the original denominators undefined.

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주요 개념

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Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including finding a common denominator and simplifying, is crucial for solving equations involving them. In this problem, the rational expressions on both sides of the equation must be combined and simplified to isolate the variable.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for simplifying rational expressions and solving equations. In the given equation, factoring the quadratic expressions in the denominators will help identify common factors and simplify the equation effectively.

Finding Common Denominators

Finding a common denominator is a key step in adding or equating rational expressions. It allows for the combination of fractions into a single expression, making it easier to solve the equation. In this case, determining the least common denominator of the fractions on both sides will facilitate the elimination of the denominators and lead to a solvable equation.