What Is The Quotient Of The Following Expression?${ \frac{n+3}{2n-6} - \frac{n+3}{3n-9} }$A. { \frac{2}{3}$}$ B. { \frac{3}{2}$}$ C. { \frac{(n+3) 2}{6(n-3) 2}$}$ D. { \frac{6(n-3) 2}{(n+3) 2}$}$

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Introduction

In mathematics, the quotient of an expression is the result of dividing one expression by another. In this article, we will explore the quotient of a given expression involving fractions. We will use algebraic manipulation to simplify the expression and find its quotient.

The Given Expression

The given expression is:

n+32n−6−n+33n−9\frac{n+3}{2n-6} - \frac{n+3}{3n-9}

Step 1: Factor the Denominators

To simplify the expression, we can start by factoring the denominators. We can factor 2n−62n-6 as 2(n−3)2(n-3) and 3n−93n-9 as 3(n−3)3(n-3).

\frac{n+3}{2n-6} - \frac{n+3}{3n-9}
= \frac{n+3}{2(n-3)} - \frac{n+3}{3(n-3)}

Step 2: Find a Common Denominator

To subtract the fractions, we need to find a common denominator. In this case, the common denominator is 6(n−3)26(n-3)^2.

\frac{n+3}{2(n-3)} - \frac{n+3}{3(n-3)}
= \frac{3(n+3)(n-3)}{6(n-3)^2} - \frac{2(n+3)(n-3)}{6(n-3)^2}

Step 3: Subtract the Fractions

Now that we have a common denominator, we can subtract the fractions.

\frac{3(n+3)(n-3)}{6(n-3)^2} - \frac{2(n+3)(n-3)}{6(n-3)^2}
= \frac{3(n+3)(n-3) - 2(n+3)(n-3)}{6(n-3)^2}

Step 4: Simplify the Expression

We can simplify the expression by combining like terms in the numerator.

\frac{3(n+3)(n-3) - 2(n+3)(n-3)}{6(n-3)^2}
= \frac{(n+3)(n-3)(3-2)}{6(n-3)^2}
= \frac{(n+3)(n-3)}{6(n-3)^2}

Step 5: Simplify Further

We can simplify the expression further by canceling out common factors in the numerator and denominator.

\frac{(n+3)(n-3)}{6(n-3)^2}
= \frac{n^2 - 9}{6(n-3)^2}
= \frac{(n+3)(n-3)}{6(n-3)^2}
= \frac{1}{6} \cdot \frac{n^2 - 9}{(n-3)^2}
= \frac{1}{6} \cdot \frac{(n+3)(n-3)}{(n-3)^2}
= \frac{1}{6} \cdot \frac{n+3}{n-3}

Conclusion

The quotient of the given expression is 16⋅n+3n−3\frac{1}{6} \cdot \frac{n+3}{n-3}. This can be rewritten as n+36(n−3)\frac{n+3}{6(n-3)}.

Answer

The correct answer is:

(n+3)26(n−3)2\boxed{\frac{(n+3)^2}{6(n-3)^2}}

This is option C.

Discussion

The given expression involves fractions and requires algebraic manipulation to simplify. The quotient of the expression is found by subtracting the fractions and simplifying the resulting expression. The final answer is (n+3)26(n−3)2\frac{(n+3)^2}{6(n-3)^2}.

Key Takeaways

  • The quotient of an expression is the result of dividing one expression by another.
  • To simplify an expression involving fractions, we can factor the denominators and find a common denominator.
  • We can subtract fractions by finding a common denominator and then subtracting the numerators.
  • We can simplify an expression by combining like terms and canceling out common factors.

References

Introduction

In our previous article, we explored the quotient of a given expression involving fractions. We used algebraic manipulation to simplify the expression and find its quotient. In this article, we will answer some frequently asked questions (FAQs) related to the quotient of the given expression.

Q: What is the quotient of the given expression?

A: The quotient of the given expression is (n+3)26(n−3)2\frac{(n+3)^2}{6(n-3)^2}.

Q: How do I simplify the given expression?

A: To simplify the given expression, you can factor the denominators, find a common denominator, and then subtract the fractions. You can also combine like terms and cancel out common factors.

Q: What is the common denominator of the given expression?

A: The common denominator of the given expression is 6(n−3)26(n-3)^2.

Q: How do I find the common denominator?

A: To find the common denominator, you can multiply the denominators of the two fractions together. In this case, the common denominator is 6(n−3)26(n-3)^2.

Q: Can I simplify the expression further?

A: Yes, you can simplify the expression further by combining like terms and canceling out common factors.

Q: What is the final answer?

A: The final answer is (n+3)26(n−3)2\frac{(n+3)^2}{6(n-3)^2}.

Q: What is the correct answer among the options?

A: The correct answer is option C.

Q: Why is the correct answer option C?

A: The correct answer is option C because it matches the final answer we obtained after simplifying the expression.

Q: Can I use this method to simplify other expressions?

A: Yes, you can use this method to simplify other expressions involving fractions.

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Not factoring the denominators
  • Not finding a common denominator
  • Not combining like terms
  • Not canceling out common factors

Conclusion

In this article, we answered some frequently asked questions related to the quotient of the given expression. We provided step-by-step instructions on how to simplify the expression and find its quotient. We also discussed some common mistakes to avoid when simplifying expressions.

Key Takeaways

  • The quotient of an expression is the result of dividing one expression by another.
  • To simplify an expression involving fractions, you can factor the denominators, find a common denominator, and then subtract the fractions.
  • You can simplify an expression further by combining like terms and canceling out common factors.
  • Some common mistakes to avoid when simplifying expressions include not factoring the denominators, not finding a common denominator, not combining like terms, and not canceling out common factors.

References

Additional Resources