Find The Quotient. Write Your Answer In Simplest Form.${ \frac{4x 2-9}{6x 2+13x+6} \div \frac{4x 2-1}{6x 2+x-2} }$(1 Point)A. { \frac{2x+3}{3x+2}$}$B. { \frac{2x-3}{2x+1}$}$C. { \frac{2x-3}{2x-5}$}$D.
Introduction
In algebra, simplifying expressions is a crucial step in solving equations and inequalities. When dealing with fractions, we often need to divide one fraction by another. This process involves inverting the second fraction and multiplying it by the first fraction. In this article, we will explore how to find the quotient of two algebraic expressions, focusing on simplifying the resulting expression.
Understanding the Problem
The given problem involves finding the quotient of two algebraic expressions:
{ \frac{4x^2-9}{6x^2+13x+6} \div \frac{4x^2-1}{6x^2+x-2} \}
To solve this problem, we need to apply the rules of division for fractions, which state that dividing one fraction by another is equivalent to multiplying the first fraction by the reciprocal of the second fraction.
Step 1: Invert the Second Fraction
The first step in finding the quotient is to invert the second fraction. This means we need to flip the numerator and denominator of the second fraction:
{ \frac{4x^2-9}{6x^2+13x+6} \div \frac{4x^2-1}{6x^2+x-2} = \frac{4x^2-9}{6x^2+13x+6} \times \frac{6x^2+x-2}{4x^2-1} \}
Step 2: Multiply the Numerators and Denominators
Next, we need to multiply the numerators and denominators of the two fractions:
{ \frac{(4x^2-9)(6x^2+x-2)}{(6x^2+13x+6)(4x^2-1)} \}
Step 3: Simplify the Expression
Now, we need to simplify the resulting expression by multiplying out the numerators and denominators:
{ \frac{24x^4+4x^3-18x^2+9x^3-9x^2-18x+18}{24x^4+52x^3+36x^2+13x+6-4x^2+1} \}
Step 4: Combine Like Terms
We can simplify the expression further by combining like terms in the numerator and denominator:
{ \frac{24x^4+9x^3-27x^2-18x+18}{24x^4+48x^3+32x^2+14x+7} \}
Step 5: Factor the Numerator and Denominator
To simplify the expression even further, we can try to factor the numerator and denominator:
{ \frac{(3x^2-1)(8x^2+9)}{(3x^2+7)(8x^2+2x+1)} \}
Step 6: Cancel Common Factors
Finally, we can cancel any common factors between the numerator and denominator:
{ \frac{(3x^2-1)(8x^2+9)}{(3x^2+7)(8x^2+2x+1)} = \frac{8x^2+9}{2x+1} \}
Conclusion
In this article, we have explored how to find the quotient of two algebraic expressions. By applying the rules of division for fractions and simplifying the resulting expression, we have arrived at the final answer:
{ \frac{8x^2+9}{2x+1} \}
This expression cannot be simplified further, and it is the simplest form of the quotient.
Discussion
The given problem is a classic example of how to find the quotient of two algebraic expressions. By following the steps outlined in this article, we have demonstrated how to simplify the resulting expression and arrive at the final answer. This problem is an excellent exercise in algebraic manipulation and simplification.
Final Answer
The final answer is:
Introduction
In our previous article, we explored how to find the quotient of two algebraic expressions. We applied the rules of division for fractions and simplified the resulting expression to arrive at the final answer. In this article, we will address some common questions and concerns related to finding the quotient of algebraic expressions.
Q: What is the first step in finding the quotient of two algebraic expressions?
A: The first step in finding the quotient of two algebraic expressions is to invert the second fraction. This means we need to flip the numerator and denominator of the second fraction.
Q: Why do we need to invert the second fraction?
A: We need to invert the second fraction because dividing one fraction by another is equivalent to multiplying the first fraction by the reciprocal of the second fraction.
Q: What is the next step after inverting the second fraction?
A: After inverting the second fraction, we need to multiply the numerators and denominators of the two fractions.
Q: Can we simplify the expression before multiplying the numerators and denominators?
A: No, we cannot simplify the expression before multiplying the numerators and denominators. We need to follow the order of operations and multiply the numerators and denominators first.
Q: How do we simplify the resulting expression after multiplying the numerators and denominators?
A: We can simplify the resulting expression by combining like terms in the numerator and denominator.
Q: Can we factor the numerator and denominator to simplify the expression further?
A: Yes, we can try to factor the numerator and denominator to simplify the expression further.
Q: What is the final step in finding the quotient of two algebraic expressions?
A: The final step in finding the quotient of two algebraic expressions is to cancel any common factors between the numerator and denominator.
Q: What if there are no common factors between the numerator and denominator?
A: If there are no common factors between the numerator and denominator, then the expression cannot be simplified further, and it is the simplest form of the quotient.
Q: Can we use the quotient rule to find the quotient of two algebraic expressions?
A: Yes, we can use the quotient rule to find the quotient of two algebraic expressions. The quotient rule states that the quotient of two functions is equal to the product of the first function and the reciprocal of the second function.
Q: What are some common mistakes to avoid when finding the quotient of two algebraic expressions?
A: Some common mistakes to avoid when finding the quotient of two algebraic expressions include:
- Not inverting the second fraction
- Not multiplying the numerators and denominators
- Not simplifying the resulting expression
- Not factoring the numerator and denominator
- Not canceling common factors between the numerator and denominator
Conclusion
In this article, we have addressed some common questions and concerns related to finding the quotient of algebraic expressions. By following the steps outlined in this article, we can ensure that we find the correct quotient and simplify the resulting expression.
Final Tips
- Make sure to follow the order of operations when finding the quotient of two algebraic expressions.
- Simplify the resulting expression by combining like terms and factoring the numerator and denominator.
- Cancel any common factors between the numerator and denominator to arrive at the final answer.
- Use the quotient rule to find the quotient of two algebraic expressions.
- Avoid common mistakes such as not inverting the second fraction, not multiplying the numerators and denominators, and not simplifying the resulting expression.
Final Answer
The final answer is: