Question

Consider the expression (3x + 6)/(x² - x - 6) + 2x/(x² + 2 - 12). Place the steps required to determine the sum of the two expressions in the correct order.

Answer 1

To determine the sum of the expressions (3x + 6)/(x² - x - 6) + 2x/(x² + 2 - 12), simplify the denominators, find a common denominator, adjust numerators accordingly, combine like terms, eliminate terms to simplify, and check the answer for correctness.

Step-by-step explanation:

To determine the sum of the two algebraic fractions (3x + 6)/(x² - x - 6) + 2x/(x² + 2 - 12), you need to follow these steps:

1. First, simplify the denominators of both fractions. For the first fraction, it's already in its lowest form, x² - x - 6. For the second fraction, simplify x² + 2 - 12 to x² - 10.
2. Next, find a common denominator for the two fractions. In this case, since the denominators are quadratic, we check if they can be factored to find common terms. Factoring the first denominator, we get (x - 3)(x + 2). The second denominator is already in its simplest form and does not have common factors with the first one. Therefore, the common denominator will be the product of the two, (x - 3)(x + 2)(x² - 10).
3. Adjust the numerators of both fractions so that both have the common denominator. Multiply the numerator and denominator of the first fraction by (x² - 10) and the second one by (x - 3)(x + 2).
4. Simplify the resulting expression by combining like terms in the numerator.
5. Eliminate terms wherever possible to simplify the algebra further.
6. Finally, check the answer to see if it is reasonable and there are no mathematical errors.