Solve each equation, if possible. (x)/(x^2-9)-(x-4)/(x^2+3 x)=(10)/(x^2-3 x)

Name: Problem 67, Chapter 1: Algebra and Trigonometry
Uploaded: 2021-06-13T09:26:18-08:00
Duration: 1 min 45 s
Channel: Yujie Wang
Description: Solve each equation, if possible. (x)/(x^2-9)-(x-4)/(x^2+3 x)=(10)/(x^2-3 x)

step 1 Okay, so we write the denominator as a product of two terms. Okay, for this one, x squared minus

Solve each equation, if possible. (x)/(x^2-9)-(x-4)/(x^2+3...

Key Concepts

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator and denominator. Solving these equations typically requires careful manipulation to combine terms, eliminate the fractions, and find the variable values that satisfy the equation while considering constraints on the domain of the rational expressions.

Domain Restrictions

When solving rational equations, it is critical to determine the domain of the expression by identifying values that cause any denominator to be zero. These restrictions must be taken into account throughout the solution process to avoid including non-permissible or extraneous solutions.

Common Denominator

Using a common denominator is a key strategy in solving rational equations. By rewriting each fraction so that they share the same denominator, one can combine terms and simplify the equation into a more manageable form, typically transforming the rational equation into a polynomial equation.

Factoring Polynomials

Factoring is an important algebraic technique used to simplify expressions and solve equations. In the context of rational equations, factoring helps identify common factors in the denominators, determine domain restrictions, and facilitate the process of clearing the denominators by allowing for cancellation or multiplication by factors.

Extraneous Solutions

Extraneous solutions are apparent solutions obtained during the process of solving an equation but are not valid for the original equation due to restrictions such as division by zero. It is essential to check all obtained solutions in the original equation to ensure that they do not violate any domain restrictions or other constraints.

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