Solve each equation. (3 a-1)/(a^2+4 a+4)-(3)/(a^2+2 a)=(3)/(a)

Name: Problem 33, Chapter 1: Precalculus
Uploaded: 2021-07-17T02:53:55-08:00
Duration: 1 min 54 s
Channel: Erika Bustos
Description: Solve each equation. (3 a-1)/(a^2+4 a+4)-(3)/(a^2+2 a)=(3)/(a)

step 1 We need to solve the following equation, but we first need to find a common denominator. So we'r

Solve each equation. (3 a-1)/(a^2+4 a+4)-(3)/(a^2+2...

Key Concepts

Rational Equations

Rational equations are equations involving fractions whose numerators and denominators are polynomials. Solving these equations typically requires manipulating and simplifying the rational expressions, while ensuring to consider the restrictions posed by the denominators. They appear in many areas of algebra, and solving them requires careful steps to prevent errors due to excluded values.

Domain Restrictions

When dealing with rational expressions, it is important to determine the domain of the variable by identifying values that make any denominator equal to zero. These restrictions must be considered during the solving process to avoid including extraneous solutions which are not valid in the original equation.

Factoring

Factoring is a fundamental algebraic technique that involves expressing a polynomial as a product of simpler polynomials. This process is key in simplifying rational expressions, especially for identifying common factors between numerators and denominators, which often facilitates the simplification and solution of equations.

Clearing Fractions

Clearing fractions is a method used to simplify rational equations by multiplying both sides by a common denominator. This eliminates the fractions and converts the equation into a polynomial equation, which is typically easier to solve. Care must be taken to ensure that the multiplication does not introduce extraneous solutions.

Solving Polynomial Equations

After clearing fractions, many rational equations transform into polynomial equations. Solving these involves using techniques such as factoring, the quadratic formula, or other algebraic methods to find the roots. These solutions are then checked against the original domain restrictions to ensure their validity.

Extraneous Solutions

Extraneous solutions are results that arise during the algebraic manipulation of an equation, often when the equation is multiplied by an expression containing the variable. Such solutions must be verified against the original equation, since they might not satisfy the original equation, especially in rational equations with domain restrictions.

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