Subtract:$ \frac{x^2 + 8x}{3x^2 + 3x} - \frac{x^2 + 2x - 6}{3x^2 + 3x} $A. $ \frac{2x - 2}{x^2 + X} $B. $ \frac{2}{x} $C. $ \frac{10x - 6}{3x^2 + 3x} $

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Introduction

In this article, we will delve into the world of algebraic expressions and learn how to subtract two fractions with the same denominator. We will start by understanding the concept of subtracting fractions, then move on to simplifying the given expression, and finally, we will compare the result with the given options to determine the correct answer.

Understanding the Concept of Subtracting Fractions

When we subtract two fractions with the same denominator, we can simply subtract the numerators and keep the denominator the same. This is because the denominator represents the total number of equal parts, and the numerator represents the number of parts we have. By subtracting the numerators, we are essentially finding the difference between the number of parts we have and the number of parts we don't have.

Simplifying the Given Expression

To simplify the given expression, we need to subtract the two fractions with the same denominator. We can do this by subtracting the numerators and keeping the denominator the same.

x2+8x3x2+3x−x2+2x−63x2+3x\frac{x^2 + 8x}{3x^2 + 3x} - \frac{x^2 + 2x - 6}{3x^2 + 3x}

=(x2+8x)−(x2+2x−6)3x2+3x= \frac{(x^2 + 8x) - (x^2 + 2x - 6)}{3x^2 + 3x}

=x2+8x−x2−2x+63x2+3x= \frac{x^2 + 8x - x^2 - 2x + 6}{3x^2 + 3x}

=6x+63x2+3x= \frac{6x + 6}{3x^2 + 3x}

Factoring the Numerator and Denominator

To simplify the expression further, we can factor the numerator and denominator.

=6(x+1)3(x2+x)= \frac{6(x + 1)}{3(x^2 + x)}

=2(x+1)(x2+x)= \frac{2(x + 1)}{(x^2 + x)}

Simplifying the Expression

We can simplify the expression by canceling out the common factor of 2 in the numerator and denominator.

=x+1(x2+x)/2= \frac{x + 1}{(x^2 + x)/2}

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

=2x= \frac{2}{x}

Comparing the Result with the Given Options

Now that we have simplified the expression, we can compare the result with the given options to determine the correct answer.

A. 2x−2x2+x\frac{2x - 2}{x^2 + x}

B. 2x\frac{2}{x}

C. 10x−63x2+3x\frac{10x - 6}{3x^2 + 3x}

The correct answer is B. 2x\frac{2}{x}.

Conclusion

In this article, we learned how to subtract two fractions with the same denominator and simplify the resulting expression. We also compared the result with the given options to determine the correct answer. The correct answer is B. 2x\frac{2}{x}.

Frequently Asked Questions

  • What is the concept of subtracting fractions?
  • How do we simplify the given expression?
  • What is the correct answer?

Final Answer

The final answer is B. 2x\frac{2}{x}.

Introduction

In our previous article, we learned how to subtract two fractions with the same denominator and simplify the resulting expression. However, we understand that there may be some questions and doubts that readers may have. In this article, we will address some of the frequently asked questions related to subtracting fractions.

Q&A

Q1: What is the concept of subtracting fractions?

A1: The concept of subtracting fractions is based on the idea of finding the difference between two or more fractions with the same denominator. When we subtract two fractions with the same denominator, we can simply subtract the numerators and keep the denominator the same.

Q2: How do we simplify the given expression?

A2: To simplify the given expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q3: What is the correct answer?

A3: The correct answer is B. 2x\frac{2}{x}.

Q4: Can we subtract fractions with different denominators?

A4: No, we cannot subtract fractions with different denominators. To subtract fractions with different denominators, we need to find the least common multiple (LCM) of the denominators and then convert both fractions to have the LCM as the denominator.

Q5: How do we find the LCM of two numbers?

A5: To find the LCM of two numbers, we can list the multiples of each number and find the smallest multiple that is common to both.

Q6: Can we add and subtract fractions with different denominators?

A6: Yes, we can add and subtract fractions with different denominators by finding the LCM of the denominators and then converting both fractions to have the LCM as the denominator.

Q7: What is the difference between adding and subtracting fractions?

A7: The main difference between adding and subtracting fractions is that when we add fractions, we add the numerators and keep the denominator the same. However, when we subtract fractions, we subtract the numerators and keep the denominator the same.

Q8: Can we simplify fractions with variables?

A8: Yes, we can simplify fractions with variables by factoring the numerator and denominator and canceling out any common factors.

Q9: How do we simplify fractions with variables?

A9: To simplify fractions with variables, we need to follow the same steps as simplifying fractions with numbers:

  1. Factor the numerator and denominator.
  2. Cancel out any common factors.
  3. Simplify the resulting expression.

Q10: Can we simplify fractions with negative numbers?

A10: Yes, we can simplify fractions with negative numbers by following the same steps as simplifying fractions with positive numbers.

Conclusion

In this article, we addressed some of the frequently asked questions related to subtracting fractions. We hope that this article has provided a clear understanding of the concept of subtracting fractions and how to simplify the resulting expression.

Final Answer

The final answer is B. 2x\frac{2}{x}.