/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 28

In the following exercises, simplify each rational expression. $$ \frac{a^{2}-5 a-36}{81-a^{2}} $$

Short Answer

Expert verified
-\frac{(a + 4)}{(9 + a)}

Step by step solution

01

- Factor the numerator

The numerator is a quadratic expression: \(a^2 - 5a - 36\). To factor it, find two numbers that multiply to -36 (the constant term) and add up to -5 (the coefficient of the linear term). These numbers are -9 and 4, so the factored form of the numerator is \((a - 9)(a + 4)\).
02

- Factor the denominator

The denominator is a difference of squares: \(81 - a^2\). Recall that \(x^2 - y^2 = (x + y)(x - y)\). Applying this to \(81 - a^2\), we get \((9 + a)(9 - a)\).
03

- Rewrite the rational expression

Substitute the factored forms of the numerator and denominator to rewrite the rational expression: \(\frac{(a - 9)(a + 4)}{(9 + a)(9 - a)}\).
04

- Simplify the expression

Notice that \(9 - a = -(a - 9)\). Thus, \((9 + a)(9 - a)\) can be rewritten using this identity. The expression \(\frac{(a - 9)(a + 4)}{(9 + a)(-(a - 9))}\) simplifies to \(-\frac{(a + 4)}{(a + 9)}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Quadratics
Factoring quadratics involves rewriting a quadratic equation in the form of ax^2 + bx + c as a product of two binomials. To do this:
  • Identify two numbers that multiply to the constant term (c) and add to the linear coefficient (b).
  • Use these numbers to break down the middle term and factor by grouping.
For example, with our quadratic expression a^2 - 5a - 36, we need numbers that multiply to -36 and add to -5. These numbers are -9 and 4, so we rewrite the expression as (a - 9)(a + 4).Breaking it down: a^2 - 9a + 4a - 36 -> a(a - 9) + 4(a - 9). The common factor here is (a - 9), leading to the final factored form.
Difference of Squares
The difference of squares formula is a handy tool in simplifying algebraic expressions. It states that x^2 - y^2 = (x + y)(x - y).This formula is applied when you have two perfect squares subtracted from each other.
In the given exercise, the denominator 81 - a^2 is a perfect example of a difference of squares. Here, 81 is 9^2, and a^2 remains as it is. Using the formula, we can factor it as: (9 + a)(9 - a), making it easier to simplify the rational expression.
Simplifying Algebraic Fractions
Simplifying algebraic fractions involves reducing the fraction to its simplest form. This process generally includes:
  • Factoring both the numerator and the denominator.
  • Cancelling out common factors.
In our exercise, once the numerator and the denominator are factored, the fraction looks like \(\frac{(a - 9)(a + 4)}{(9 + a)(9 - a)}\). Notice that 9 - a can be rewritten as -(a - 9), utilizing the identity. So, the fraction adjusts to \(\frac{(a - 9)(a + 4)}{(9 + a)(-(a - 9))}\). The term (a - 9) gets canceled, resulting in the simplified form: -\frac{(a + 4)}{(a + 9)}.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve the application problem provided. Dana enjoys taking her dog for a walk, but sometimes her dog gets away, and she has to run after him. Dana walked her dog for 7 miles but then had to run for 1 mile, spending a total time of 2.5 hours with her dog. Her running speed was 3 mph faster than her walking speed. Find her walking speed.

Solve each rational equation. $$\frac{p}{p+7}-8=\frac{98}{p^{2}-49}$$

A car's value varies inversely with its age. Jackie bought \(a\) 10-year-old car for $$\$ 2,400$$. (a) Write the equation that relates the car's value to its age. (b) What will be the value of Jackie's car when it is 15 years old?

Sonya drinks a 32-ounce energy drink containing 80 calories per 12 ounce. How many calories did she drink?

Solve each rational equation. $$\frac{2}{x^{2}+2 x-8}-\frac{1}{x^{2}+9 x+20}=\frac{4}{x^{2}+3 x-10}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.