Simplify each complex fraction. 6x2−25+x1x−5\frac{\frac{6}{$$x^2 - 25$$} + x}{\frac{1}{x - 5}} x−51x2−256+x

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 82

1장, 문제 82

Simplify each complex fraction. $$\frac{\frac{6}{x^2 - 25}$ + x}{$\frac{1}{x - 5}$}$

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Identify the complex fraction: the numerator is $ \frac{6}{x^{2} - 25} + x $ and the denominator is $ \frac{1}{x - 5} $.

Factor the quadratic expression in the denominator of the numerator: $ x^{2} - 25 $ is a difference of squares, so $ x^{2} - 25 = (x - 5)(x + 5) $. Rewrite the numerator as $ \frac{6}{(x - 5)(x + 5)} + x $.

To combine the terms in the numerator, express $ x $ as a fraction with denominator $ (x - 5)(x + 5) $, so rewrite $ x = \frac{x (x - 5)(x + 5)}{(x - 5)(x + 5)} $.

Add the two fractions in the numerator: $ \frac{6}{(x - 5)(x + 5)} + \frac{x (x - 5)(x + 5)}{(x - 5)(x + 5)} = \frac{6 + x (x - 5)(x + 5)}{(x - 5)(x + 5)} $.

Rewrite the entire complex fraction as a division: $ \frac{\frac{6 + x (x - 5)(x + 5)}{(x - 5)(x + 5)}}{\frac{1}{x - 5}} = \frac{6 + x (x - 5)(x + 5)}{(x - 5)(x + 5)} \times \frac{x - 5}{1} $. Then simplify by canceling common factors and simplifying the numerator expression.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by finding common denominators or multiplying by the reciprocal of the denominator.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of simpler binomials. For example, x^2 - 25 is a difference of squares and factors into (x - 5)(x + 5), which helps simplify expressions and cancel common factors.

Operations with Rational Expressions

Rational expressions are fractions with polynomials in numerator and denominator. Adding, subtracting, multiplying, or dividing them requires finding common denominators, factoring, and simplifying by canceling common factors to reduce the expression.

Simplify each complex fraction. 6x2−25+x1x−5\(\frac{\frac{6}{x^2 - 25}\) + x}{\(\frac{1}{x - 5}\)} x−51x2−256+x

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Complex Fractions

Factoring Quadratic Expressions

Operations with Rational Expressions

Simplify each complex fraction. $\(\frac{\frac{6}{x^2 - 25}\) + x}{\(\frac{1}{x - 5}\)}$