Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To simplify the given quotient [tex]\(\frac{3x^2 - 27x}{2x^2 + 13x - 7} \div \frac{3x}{4x^2 - 1}\)[/tex], follow these steps:
1. Rewrite the Given Expression: Convert the division problem into a multiplication problem by multiplying by the reciprocal of the second fraction.
[tex]\[ \frac{3x^2 - 27x}{2x^2 + 13x - 7} \div \frac{3x}{4x^2 - 1} = \frac{3x^2 - 27x}{2x^2 + 13x - 7} \times \frac{4x^2 - 1}{3x} \][/tex]
2. Factor the Expressions: Factorize the numerator and the denominator wherever possible.
- [tex]\(3x^2 - 27x = 3x(x - 9)\)[/tex]
- [tex]\(4x^2 - 1 = (2x - 1)(2x + 1)\)[/tex]
The middle expression [tex]\(2x^2 + 13x - 7\)[/tex] does not factor neatly, so keep it as is for now.
3. Write the Factored Form: Substitute the factors back into the expression.
[tex]\[ \frac{3x(x - 9)}{2x^2 + 13x - 7} \times \frac{(2x - 1)(2x + 1)}{3x} \][/tex]
4. Cancel Common Terms: Identify and cancel out any common terms from the numerator and the denominator.
The factor [tex]\(3x\)[/tex] appears in both the numerator and the denominator and can be cancelled out.
[tex]\[ \frac{(x - 9)(2x - 1)(2x + 1)}{2x^2 + 13x - 7} \][/tex]
Now the simplified form of the quotient is:
[tex]\[ \frac{(x - 9)(2x - 1)(2x + 1)}{2x^2 + 13x - 7} \][/tex]
5. Determine when the Expression Does Not Exist: The expression does not exist where the denominator is zero. Therefore, solve:
[tex]\[ 2x^2 + 13x - 7 = 0 \][/tex]
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 2\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = -7\)[/tex]:
The roots are [tex]\(x = 0.5\)[/tex] and [tex]\(x = -7.0\)[/tex].
6. Inclusion of [tex]\(x = 0\)[/tex]: Since [tex]\(x = 0\)[/tex] also makes the original denominator [tex]\(3x\)[/tex] in the second fraction equal to zero, we must include this in the restrictions.
Hence, the simplest form of the numerator is [tex]\((x - 9)(2x - 1)(2x + 1)\)[/tex], and the expression does not exist when [tex]\(x = 0\)[/tex], [tex]\(x = 0.5\)[/tex], or [tex]\(x = -7.0\)[/tex].
So, select the correct answer from each drop-down menu:
- The simplest form of this quotient has a numerator of [tex]\((x - 9)(2x - 1)(2x + 1)\)[/tex]
- The expression does not exist when [tex]\(x = 0\)[/tex].
1. Rewrite the Given Expression: Convert the division problem into a multiplication problem by multiplying by the reciprocal of the second fraction.
[tex]\[ \frac{3x^2 - 27x}{2x^2 + 13x - 7} \div \frac{3x}{4x^2 - 1} = \frac{3x^2 - 27x}{2x^2 + 13x - 7} \times \frac{4x^2 - 1}{3x} \][/tex]
2. Factor the Expressions: Factorize the numerator and the denominator wherever possible.
- [tex]\(3x^2 - 27x = 3x(x - 9)\)[/tex]
- [tex]\(4x^2 - 1 = (2x - 1)(2x + 1)\)[/tex]
The middle expression [tex]\(2x^2 + 13x - 7\)[/tex] does not factor neatly, so keep it as is for now.
3. Write the Factored Form: Substitute the factors back into the expression.
[tex]\[ \frac{3x(x - 9)}{2x^2 + 13x - 7} \times \frac{(2x - 1)(2x + 1)}{3x} \][/tex]
4. Cancel Common Terms: Identify and cancel out any common terms from the numerator and the denominator.
The factor [tex]\(3x\)[/tex] appears in both the numerator and the denominator and can be cancelled out.
[tex]\[ \frac{(x - 9)(2x - 1)(2x + 1)}{2x^2 + 13x - 7} \][/tex]
Now the simplified form of the quotient is:
[tex]\[ \frac{(x - 9)(2x - 1)(2x + 1)}{2x^2 + 13x - 7} \][/tex]
5. Determine when the Expression Does Not Exist: The expression does not exist where the denominator is zero. Therefore, solve:
[tex]\[ 2x^2 + 13x - 7 = 0 \][/tex]
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 2\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = -7\)[/tex]:
The roots are [tex]\(x = 0.5\)[/tex] and [tex]\(x = -7.0\)[/tex].
6. Inclusion of [tex]\(x = 0\)[/tex]: Since [tex]\(x = 0\)[/tex] also makes the original denominator [tex]\(3x\)[/tex] in the second fraction equal to zero, we must include this in the restrictions.
Hence, the simplest form of the numerator is [tex]\((x - 9)(2x - 1)(2x + 1)\)[/tex], and the expression does not exist when [tex]\(x = 0\)[/tex], [tex]\(x = 0.5\)[/tex], or [tex]\(x = -7.0\)[/tex].
So, select the correct answer from each drop-down menu:
- The simplest form of this quotient has a numerator of [tex]\((x - 9)(2x - 1)(2x + 1)\)[/tex]
- The expression does not exist when [tex]\(x = 0\)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
