What Is The Quotient?${ \frac{2y^2 - 6y - 20}{4y + 12} - \frac{y^2 + 5y + 6}{3y^2 + 18y + 27} }$A. { \frac{2}{3(y-5)}$}$B. { \frac{3(y-5)}{2}$}$C. { \frac{(y-5)(y+2) 2}{6(y+3) 2}$}$D.

Understanding the Concept of Quotient in Mathematics

In mathematics, a quotient is the result of a division operation. It is a value that represents the amount of one quantity divided by another. The quotient can be a whole number, a fraction, or a decimal. In this article, we will explore the concept of quotient and how to find it in a given expression.

The Quotient of Two Rational Expressions

A rational expression is a fraction that contains variables and constants in the numerator and denominator. To find the quotient of two rational expressions, we need to follow a specific set of steps. The given expression is:

${ \frac{2y^2 - 6y - 20}{4y + 12} - \frac{y^2 + 5y + 6}{3y^2 + 18y + 27} }$

Step 1: Factor the Numerators and Denominators

To simplify the expression, we need to factor the numerators and denominators of both fractions.

${ \frac{2y^2 - 6y - 20}{4y + 12} = \frac{2(y^2 - 3y - 10)}{4(y + 3)} }$

${ \frac{y^2 + 5y + 6}{3y^2 + 18y + 27} = \frac{(y + 2)(y + 3)}{3(y + 3)^2} }$

Step 2: Simplify the Expression

Now that we have factored the numerators and denominators, we can simplify the expression by canceling out any common factors.

${ \frac{2(y^2 - 3y - 10)}{4(y + 3)} - \frac{(y + 2)(y + 3)}{3(y + 3)^2} }$

${ = \frac{2(y - 5)(y + 2)}{4(y + 3)} - \frac{(y + 2)}{3(y + 3)} }$

Step 3: Find the Common Denominator

To subtract the fractions, we need to find a common denominator. In this case, the common denominator is 12(y + 3).

${ = \frac{6(y - 5)(y + 2)}{12(y + 3)} - \frac{4(y + 2)}{12(y + 3)} }$

Step 4: Subtract the Fractions

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

${ = \frac{6(y - 5)(y + 2) - 4(y + 2)}{12(y + 3)} }$

${ = \frac{6(y^2 - 3y - 10) - 4(y + 2)}{12(y + 3)} }$

${ = \frac{6y^2 - 18y - 60 - 4y - 8}{12(y + 3)} }$

${ = \frac{6y^2 - 22y - 68}{12(y + 3)} }$

Step 5: Factor the Numerator

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

${ = \frac{2(3y^2 - 11y - 34)}{12(y + 3)} }$

${ = \frac{(y - 5)(3y + 2)(y + 2)}{6(y + 3)^2} }$

The Final Answer

The final answer is:

${ \frac{(y - 5)(y + 2)^2}{6(y + 3)^2} }$

This is the quotient of the given expression. It is a rational expression that represents the result of dividing one quantity by another.

Conclusion

Frequently Asked Questions About Quotient

In this article, we will answer some of the most frequently asked questions about quotient. Whether you are a student, a teacher, or just someone who wants to learn more about mathematics, this article is for you.

Q: What is a quotient?

A: A quotient is the result of a division operation. It is a value that represents the amount of one quantity divided by another.

Q: How do I find the quotient of two rational expressions?

A: To find the quotient of two rational expressions, you need to follow a specific set of steps. First, factor the numerators and denominators of both fractions. Then, simplify the expression by canceling out any common factors. Next, find a common denominator and subtract the fractions. Finally, factor the numerator and simplify the expression.

Q: What is the difference between a quotient and a dividend?

A: A dividend is the number being divided, while a quotient is the result of the division operation. For example, in the expression 12 ÷ 3, 12 is the dividend and 4 is the quotient.

Q: Can a quotient be a whole number?

A: Yes, a quotient can be a whole number. For example, in the expression 12 ÷ 3, the quotient is 4, which is a whole number.

Q: Can a quotient be a fraction?

A: Yes, a quotient can be a fraction. For example, in the expression 1/2 ÷ 3/4, the quotient is 2/3, which is a fraction.

Q: Can a quotient be a decimal?

A: Yes, a quotient can be a decimal. For example, in the expression 1.5 ÷ 2, the quotient is 0.75, which is a decimal.

Q: How do I simplify a quotient?

A: To simplify a quotient, you need to factor the numerator and denominator and cancel out any common factors. You can also use the rules of exponents to simplify the expression.

Q: What is the order of operations for quotient?

A: The order of operations for quotient is:

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: Evaluate any addition and subtraction operations from left to right.

Q: Can I use a calculator to find the quotient?

A: Yes, you can use a calculator to find the quotient. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Conclusion

In this article, we answered some of the most frequently asked questions about quotient. We covered topics such as what a quotient is, how to find the quotient of two rational expressions, and how to simplify a quotient. We also discussed the order of operations for quotient and whether you can use a calculator to find the quotient. Whether you are a student, a teacher, or just someone who wants to learn more about mathematics, this article is for you.

Additional Resources

If you want to learn more about quotient, here are some additional resources you can use:

• Khan Academy: Quotient
• Mathway: Quotient
• Wolfram Alpha: Quotient
• MIT OpenCourseWare: Quotient

Final Thoughts

Quotient is an important concept in mathematics that can be used to solve a wide range of problems. By understanding how to find the quotient of two rational expressions and how to simplify a quotient, you can become a more confident and proficient mathematician. Remember to always follow the order of operations and to check your work by hand to make sure you get the correct answer.