Question

$\frac{3 a-1}{a^{2}+4 a+4}-\frac{3}{a^{2}+2 a}=\frac{3}{a}$

   $\frac{3 a-1}{a^{2}+4 a+4}-\frac{3}{a^{2}+2 a}=\frac{3}{a}$

College Algebra

College Algebra

Raymond A. Barnett,… 9th Edition

Chapter 1, Problem 33

Instant Answer

Solved by Expert Alison Rodriguez

02/25/2023

Step 1

The left-hand side is given by: \[ \frac{3 a-1}{a^{2}+4 a+4}-\frac{3}{a^{2}+2 a} \] We can factor the denominators. The first denominator can be factored as: \[ a^{2}+4 a+4 = (a+2)^2 \] The second denominator can be factored as: \[ a^{2}+2 a = a(a+2) \] Thus, we Show more…

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$\frac{3 a-1}{a^{2}+4 a+4}-\frac{3}{a^{2}+2 a}=\frac{3}{a}$

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Key Concepts

Domain Restrictions in Rational Equations

When dealing with rational expressions, it is critical to determine the values for which the expression is undefined. This involves setting each denominator equal to zero and solving for the variable, ensuring that any solutions found do not violate these restrictions.

Clearing Denominators and Solving Equations

After determining a common denominator, one can eliminate the fractional components by multiplying the entire equation by this denominator. This operation transforms the rational equation into a polynomial equation that can be solved using standard algebraic techniques.

Factoring Quadratic Expressions

This concept involves rewriting quadratic expressions as products of simpler binomials, which simplifies further manipulation. It is essential when working with rational expressions since factoring assists in identifying common factors between numerators and denominators.

Common Denominator and Rational Expression Manipulation

Combining rational expressions requires rewriting the fractions so they share a common denominator. This process entails factoring the denominators and finding the least common multiple, which then enables addition or subtraction of the fractions.

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