Perform The Indicated Operations And Simplify Each Result.1. $\frac{2}{x+2} - \frac{1}{x+10}$2. $\frac{9}{x+2} + \frac{3}{x^2+5x+6}$

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Introduction

Complex fractions are a type of mathematical expression that involves fractions within fractions. They can be challenging to simplify, but with the right approach, they can be broken down into simpler expressions. In this article, we will explore two complex fractions and provide step-by-step instructions on how to simplify each result.

Simplifying the First Complex Fraction

The first complex fraction is 2x+2−1x+10\frac{2}{x+2} - \frac{1}{x+10}. To simplify this expression, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators, which is (x+2)(x+10)(x+2)(x+10).

Step 1: Find the Common Denominator

The common denominator is (x+2)(x+10)(x+2)(x+10).

Step 2: Rewrite Each Fraction with the Common Denominator

2x+2=2(x+10)(x+2)(x+10)\frac{2}{x+2} = \frac{2(x+10)}{(x+2)(x+10)}

1x+10=1(x+2)(x+2)(x+10)\frac{1}{x+10} = \frac{1(x+2)}{(x+2)(x+10)}

Step 3: Subtract the Two Fractions

2(x+10)(x+2)(x+10)−1(x+2)(x+2)(x+10)\frac{2(x+10)}{(x+2)(x+10)} - \frac{1(x+2)}{(x+2)(x+10)}

=2(x+10)−1(x+2)(x+2)(x+10)= \frac{2(x+10) - 1(x+2)}{(x+2)(x+10)}

=2x+20−x−2(x+2)(x+10)= \frac{2x + 20 - x - 2}{(x+2)(x+10)}

=x+18(x+2)(x+10)= \frac{x + 18}{(x+2)(x+10)}

Step 4: Simplify the Expression

The expression x+18(x+2)(x+10)\frac{x + 18}{(x+2)(x+10)} is already simplified.

Simplifying the Second Complex Fraction

The second complex fraction is 9x+2+3x2+5x+6\frac{9}{x+2} + \frac{3}{x^2+5x+6}. To simplify this expression, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators, which is (x+2)(x2+5x+6)(x+2)(x^2+5x+6).

Step 1: Find the Common Denominator

The common denominator is (x+2)(x2+5x+6)(x+2)(x^2+5x+6).

Step 2: Rewrite Each Fraction with the Common Denominator

9x+2=9(x2+5x+6)(x+2)(x2+5x+6)\frac{9}{x+2} = \frac{9(x^2+5x+6)}{(x+2)(x^2+5x+6)}

3x2+5x+6=3(x+2)(x+2)(x2+5x+6)\frac{3}{x^2+5x+6} = \frac{3(x+2)}{(x+2)(x^2+5x+6)}

Step 3: Add the Two Fractions

9(x2+5x+6)(x+2)(x2+5x+6)+3(x+2)(x+2)(x2+5x+6)\frac{9(x^2+5x+6)}{(x+2)(x^2+5x+6)} + \frac{3(x+2)}{(x+2)(x^2+5x+6)}

=9(x2+5x+6)+3(x+2)(x+2)(x2+5x+6)= \frac{9(x^2+5x+6) + 3(x+2)}{(x+2)(x^2+5x+6)}

=9x2+45x+54+3x+6(x+2)(x2+5x+6)= \frac{9x^2 + 45x + 54 + 3x + 6}{(x+2)(x^2+5x+6)}

=9x2+48x+60(x+2)(x2+5x+6)= \frac{9x^2 + 48x + 60}{(x+2)(x^2+5x+6)}

Step 4: Factor the Numerator

The numerator 9x2+48x+609x^2 + 48x + 60 can be factored as 3(3x2+16x+20)3(3x^2 + 16x + 20).

Step 5: Simplify the Expression

The expression 3(3x2+16x+20)(x+2)(x2+5x+6)\frac{3(3x^2 + 16x + 20)}{(x+2)(x^2+5x+6)} is already simplified.

Conclusion

Simplifying complex fractions requires finding a common denominator and rewriting each fraction with the common denominator. Once the fractions have the same denominator, they can be added or subtracted by combining the numerators. In this article, we simplified two complex fractions and provided step-by-step instructions on how to simplify each result. By following these steps, you can simplify complex fractions and make them easier to work with.

Common Mistakes to Avoid

When simplifying complex fractions, there are several common mistakes to avoid:

  • Not finding the common denominator: Failing to find the common denominator can make it difficult to simplify the expression.
  • Not rewriting each fraction with the common denominator: Failing to rewrite each fraction with the common denominator can make it difficult to add or subtract the fractions.
  • Not combining the numerators: Failing to combine the numerators can make it difficult to simplify the expression.

Tips for Simplifying Complex Fractions

When simplifying complex fractions, here are some tips to keep in mind:

  • Find the least common multiple (LCM) of the denominators: The LCM is the smallest multiple that both denominators have in common.
  • Rewrite each fraction with the common denominator: This will make it easier to add or subtract the fractions.
  • Combine the numerators: This will make it easier to simplify the expression.
  • Factor the numerator: Factoring the numerator can make it easier to simplify the expression.

Introduction

Simplifying complex fractions can be a challenging task, but with the right approach, it can be broken down into simpler expressions. In this article, we will provide a Q&A guide to help you simplify complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a type of mathematical expression that involves fractions within fractions. It is a fraction that contains one or more fractions in the numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to find a common denominator and rewrite each fraction with the common denominator. Once the fractions have the same denominator, you can add or subtract the fractions by combining the numerators.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the two denominators. It is the smallest multiple that both denominators have in common.

Q: How do I find the common denominator?

A: To find the common denominator, you need to list the multiples of each denominator and find the smallest multiple that both denominators have in common.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.

Q: How do I rewrite each fraction with the common denominator?

A: To rewrite each fraction with the common denominator, you need to multiply the numerator and denominator of each fraction by the necessary factors to make the denominators equal.

Q: How do I combine the numerators?

A: To combine the numerators, you need to add or subtract the numerators of the two fractions, depending on whether you are adding or subtracting the fractions.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

  • Not finding the common denominator
  • Not rewriting each fraction with the common denominator
  • Not combining the numerators
  • Not factoring the numerator

Q: What are some tips for simplifying complex fractions?

A: Some tips for simplifying complex fractions include:

  • Finding the least common multiple (LCM) of the denominators
  • Rewriting each fraction with the common denominator
  • Combining the numerators
  • Factoring the numerator

Q: Can you provide an example of simplifying a complex fraction?

A: Yes, here is an example of simplifying a complex fraction:

2x+2−1x+10\frac{2}{x+2} - \frac{1}{x+10}

To simplify this expression, we need to find a common denominator, which is (x+2)(x+10)(x+2)(x+10). We then rewrite each fraction with the common denominator:

2(x+10)(x+2)(x+10)−1(x+2)(x+2)(x+10)\frac{2(x+10)}{(x+2)(x+10)} - \frac{1(x+2)}{(x+2)(x+10)}

We then combine the numerators:

2(x+10)−1(x+2)(x+2)(x+10)\frac{2(x+10) - 1(x+2)}{(x+2)(x+10)}

We then simplify the expression:

x+18(x+2)(x+10)\frac{x + 18}{(x+2)(x+10)}

Conclusion

Simplifying complex fractions requires finding a common denominator and rewriting each fraction with the common denominator. Once the fractions have the same denominator, you can add or subtract the fractions by combining the numerators. By following these steps and avoiding common mistakes, you can simplify complex fractions and make them easier to work with.

Common Complex Fraction Problems

Here are some common complex fraction problems:

  • 2x+2+3x2+5x+6\frac{2}{x+2} + \frac{3}{x^2+5x+6}
  • 9x+2−1x+10\frac{9}{x+2} - \frac{1}{x+10}
  • 4x2+4x+4−2x+2\frac{4}{x^2+4x+4} - \frac{2}{x+2}
  • 3x+2+2x2+5x+6\frac{3}{x+2} + \frac{2}{x^2+5x+6}

Practice Problems

Here are some practice problems to help you simplify complex fractions:

  • 2x+2−1x+10\frac{2}{x+2} - \frac{1}{x+10}
  • 9x+2+3x2+5x+6\frac{9}{x+2} + \frac{3}{x^2+5x+6}
  • 4x2+4x+4−2x+2\frac{4}{x^2+4x+4} - \frac{2}{x+2}
  • 3x+2+2x2+5x+6\frac{3}{x+2} + \frac{2}{x^2+5x+6}

By practicing these problems and following the steps outlined in this article, you can become more confident in your ability to simplify complex fractions.