/*! 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 51 add or subtract as indicated. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 51

add or subtract as indicated. $$ \frac{4}{x^{2}+6 x+9}+\frac{4}{x+3} $$

Short Answer

Expert verified
The result of the addition is \( \frac{4x+16}{(x+3)^{2}} \)

Step by step solution

01

Identify the common denominator

Noticing that the denominator \( x^{2}+6x+9 \) is the perfect square of the binomial \( x+3 \), which is the denominator of the second fraction. So this perfect square, \( x^{2}+6x+9 = (x+3)^{2} \), is the common denominator.
02

Transform the second fraction to have the common denominator

The denominator \( x+3 \) can be rewritten as \( (x+3)^{2} \) when multiplied by \( x+3 \). So the fraction becomes \( \frac{4*(x+3)}{(x+3)^{2}} = \frac{4x+12}{(x+3)^{2}} \)
03

Add fractions with the common denominator

Now both fractions have the same denominator \( (x+3)^{2} \), they can be added together: \( \frac{4x+12}{(x+3)^{2}} + \frac{4}{(x+3)^{2}} = \frac{4x+16}{(x+3)^{2}} \)

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 Denominator
When dealing with algebraic fractions, a common denominator is essential for addition or subtraction. It harmonizes the fractions by bringing both under the same baseline. In our exercise, we need a common denominator to add the fractions \( \frac{4}{x^2+6x+9} \) and \( \frac{4}{x+3} \). A keen observation reveals that the denominator \( x^2+6x+9 \) is a perfect square of the binomial \( x+3 \). Thus, we can rewrite it as \((x+3)^2\). Using this insight, we understand that the common denominator for both fractions is \((x+3)^2\).
  • This simplifies the process, making it easier to add the fractions.
  • Recognizing perfect squares like \((x+3)^2\) helps identify common denominators quickly.
A shared denominator allows us to combine the fractions effortlessly.
Adding Fractions
Adding fractions involves bringing them under a single denominator to integrate the numerators. In this exercise, after identifying the common denominator \((x+3)^2\), we need to transform the second fraction. Initially, the fraction \(\frac{4}{x+3}\) must be adapted to share the common denominator. This requires multiplying the numerator and the denominator by \(x+3\), giving us \(\frac{4(x+3)}{(x+3)^2} = \frac{4x+12}{(x+3)^2}\).
  • This step ensures both fractions have compatible structures for addition.
  • Multiplying keeps the fraction value equivalent when upping the denominator.
With this setup, the fractions become \( \frac{4x+12}{(x+3)^2} \) and \( \frac{4}{(x+3)^2} \). Adding these results in \( \frac{4x+16}{(x+3)^2} \). This math relies on unified denominators to sum numerators smoothly.
Perfect Square
Perfect squares simplify many algebra problems by representing polynomials in a neat, factorable form. For instance, the polynomial \(x^2+6x+9\) is a perfect square since it can be expressed as \((x+3)^2\). Recognizing this helps streamline operations, such as finding a common denominator in fraction addition.
  • Perfect squares often arise from expressions like \((a+b)^2\), where the middle term is double the product of the roots \((2ab)\).
  • This characteristic aids in reducing polynomial expressions and finding factors.
In our exercise, realizing the expression as a perfect square allowed efficient handling and combination of the fractions. Appreciating these forms can greatly benefit algebraic manipulations.

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

In parts (a) and (b), complete each statement. $$ \text { a. } \frac{b^{7}}{b^{3}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}=b^{?} $$ $$ \text { b. } \frac{b^{8}}{b^{2}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}=b^{?} $$ c. Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ (-2)^{4}=2^{-4} $$

Factor Completely. $$(y+1)^{3}+1$$

What does it mean when we say that a formula models real-world phenomena?

Use Einstein's special-relativity equation $$R_{a}=R_{f} \sqrt{1-\left(\frac{v}{c}\right)^{2}}$$ described in the Blitzer Bonus on page 47,to solve this exercise. You are moving at \(90 \%\) of the speed of light. Substitute 0.9c for v, your velocity, in the equation. What is your aging rate, correct to two decimal places, relative to a friend on Earth? If you are gone for 44 weeks, approximately how many weeks have passed for your friend?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.