/*! 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 36 For exercises \(5-48\), simplify... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 36

For exercises \(5-48\), simplify. $$ \frac{2 w^{2}}{2 w^{2}-11 w-6}-\frac{-5 w-2}{2 w^{2}-11 w-6} $$

Short Answer

Expert verified
\( \frac{2w^2 + 5w + 2}{2w^2 - 11w - 6} \)

Step by step solution

01

Identify Common Denominator

Observe that both fractions have the same denominator: \(2w^2 - 11w - 6\).
02

Combine the Numerators

Since the denominators are the same, subtract the numerators: \[ \frac{2w^2}{2w^2 - 11w - 6} - \frac{-5w - 2}{2w^2 - 11w - 6} = \frac{2w^2 - (-5w - 2)}{2w^2 - 11w - 6} \]
03

Simplify the Numerator

Distribute the negative sign and combine like terms: \[ 2w^2 - (-5w - 2) = 2w^2 + 5w + 2 \]
04

Rewrite the Fraction

Rewrite the fraction with the simplified numerator: \[ \frac{2w^2 + 5w + 2}{2w^2 - 11w - 6} \]
05

Final Simplification

The fraction \( \frac{2w^2 + 5w + 2}{2w^2 - 11w - 6} \) is in its simplest form as the numerator and the denominator do not have common factors.

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.

common denominators
When simplifying algebraic fractions, it is important to identify if the denominators are the same. If they are, you can combine the fractions directly. In this exercise, both fractions have the same denominator: \(2w^2 - 11w - 6\). This makes it straightforward to subtract the fractions because the computation only involves the numerators. Knowing the denominators are identical allows you to focus on simplifying the numerators first. This step sets the stage for combining and simplifying further.
combining like terms
After making sure the denominators are the same, the next step is to combine the numerators. In this case, we subtract the second numerator from the first: \( \frac{2w^2}{2w^2 - 11w - 6} - \frac{-5w - 2}{2w^2 - 11w - 6} \). Once you align them over the common denominator, this process is easier. The numerators are now combined: \(2w^2 - (-5w - 2)\). Combining like terms means adding or subtracting the same variables or constants together. This includes aligning the terms properly and simplifying them into a single expression.
distributing negative signs
Distributing negative signs is crucial for simplifying expressions correctly. In the given exercise, you need to subtract the second fraction's numerator from the first. Here, it's important to distribute the negative sign correctly: \(2w^2 - (-5w - 2)\). When you distribute the negative sign, it changes the signs of the terms inside the parentheses: \(2w^2 + 5w + 2\). Always ensure every term inside the parentheses is correctly affected by the negative sign. This helps prevent errors in further simplifications.
simplifying expressions
After distributing and combining like terms, the final focus is on simplifying the expression. In this exercise, you reached \( \frac{2w^2 + 5w + 2}{2w^2 - 11w - 6} \). Check if the numerator and the denominator share any common factors. If they don't, the fraction is in its simplest form. Here, since \(2w^2 + 5w + 2\) and \(2w^2 - 11w - 6\) have no common factors, the fraction cannot be simplified further. A simplified expression means no further reduction is possible, making it easier to understand and use in other calculations.

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

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. The water from a garden hose turned on at full pressure fills a hot tub in \(45 \mathrm{~min}\). If the drain is open, the hot tub empties in \(62 \mathrm{~min}\). Find the amount of time to fill the hot tub with the drain open. Round to the nearest whole number.

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)

For exercises 1-10, (a) solve. (b) check. $$ \frac{4}{9} p-\frac{1}{8}=\frac{25}{72} $$

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{3} x+\frac{3}{2}=\frac{1}{3} x+\frac{1}{6} $$

The rulebook of the U.S. Lawn Mower Racing Association describes how to award points. 100 points for registration 100 points for starting a race 100 points for finishing a race 300 points for first place 250 points for second place 200 points for third place 150 points for fourth place 100 points for fifth place Source: www.letsmow.com A lawn mower racer registered for a day of racing. She started and completed three races. She placed fourth in the first race, third in the second race, and first in the final race. Find the total number of points she earned.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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.