Question
Identify the equivalent expression in the equation 1/x^2−x+1/x=5/x^2−x
and demonstrate multiplying by the common denominator.(1 point)
Responses
x^2(1/x^2−x)+x^2(1/x)=x^2(5/x^2−x)
x squared left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus x squared left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals x squared left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
(x^2−x)(1/x^2−x)+(x^2−x)(1/x)=(x^2−x)(5/x^2−x)
left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
x(1/x^2−x)+x(1/x)=x(5/x^2−x)
x left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus x left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals x left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
(x−1)(1/x^2−x)+(x−1)(1/x)=(x−1)(5/x^2−x)
left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
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and demonstrate multiplying by the common denominator.(1 point)
Responses
x^2(1/x^2−x)+x^2(1/x)=x^2(5/x^2−x)
x squared left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus x squared left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals x squared left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
(x^2−x)(1/x^2−x)+(x^2−x)(1/x)=(x^2−x)(5/x^2−x)
left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals left parenthesis x squared minus x right parenthesis left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
x(1/x^2−x)+x(1/x)=x(5/x^2−x)
x left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus x left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals x left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
(x−1)(1/x^2−x)+(x−1)(1/x)=(x−1)(5/x^2−x)
left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 1 over x squared minus x End Fraction right parenthesis plus left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis Start Fraction 5 over x squared minus x End Fraction right parenthesis
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Answers
Answered by
GPT-4o mini
To simplify the equation \( \frac{1}{x^2} - x + \frac{1}{x} = \frac{5}{x^2} - x \) by multiplying by a common denominator, we can identify the common denominator first. The denominators present in the equation are \( x^2 \) and \( x \). The least common denominator (LCD) in this case is \( x^2 \).
We will multiply each term of the equation by the common denominator \( x^2 \):
1. The left side of the equation:
- Multiply \( \frac{1}{x^2} \) by \( x^2 \): \( 1 \)
- Multiply \( -x \) by \( x^2 \): \( -x^3 \)
- Multiply \( \frac{1}{x} \) by \( x^2 \): \( x \)
Thus, the left side becomes:
\[
x^2\left(\frac{1}{x^2}\right) - x^2(x) + x^2\left(\frac{1}{x}\right) = 1 - x^3 + x
\]
2. The right side of the equation:
- Multiply \( \frac{5}{x^2} \) by \( x^2 \): \( 5 \)
- Multiply \( -x \) by \( x^2 \): \( -x^3 \)
Thus, the right side becomes:
\[
x^2\left(\frac{5}{x^2}\right) - x^2(x) = 5 - x^3
\]
Putting it all together, we have:
\[
1 - x^3 + x = 5 - x^3
\]
Now, matching this with potential responses, the expression derived from the common denominator approach is:
\[
1 - x^3 + x = 5 - x^3
\]
Among the given choices, the correct one representing the equivalent expression after multiplying by the common denominator \( x^2 \) is:
\[
x^2\left(\frac{1}{x^2} - x\right) + x^2\left(\frac{1}{x}\right) = x^2\left(\frac{5}{x^2} - x\right)
\]
Thus, the closest response would be:
\[
x^2\left( \frac{1}{x^2} - x \right) + x^2\left( \frac{1}{x} \right) = x^2\left( \frac{5}{x^2} - x \right)
\]
We will multiply each term of the equation by the common denominator \( x^2 \):
1. The left side of the equation:
- Multiply \( \frac{1}{x^2} \) by \( x^2 \): \( 1 \)
- Multiply \( -x \) by \( x^2 \): \( -x^3 \)
- Multiply \( \frac{1}{x} \) by \( x^2 \): \( x \)
Thus, the left side becomes:
\[
x^2\left(\frac{1}{x^2}\right) - x^2(x) + x^2\left(\frac{1}{x}\right) = 1 - x^3 + x
\]
2. The right side of the equation:
- Multiply \( \frac{5}{x^2} \) by \( x^2 \): \( 5 \)
- Multiply \( -x \) by \( x^2 \): \( -x^3 \)
Thus, the right side becomes:
\[
x^2\left(\frac{5}{x^2}\right) - x^2(x) = 5 - x^3
\]
Putting it all together, we have:
\[
1 - x^3 + x = 5 - x^3
\]
Now, matching this with potential responses, the expression derived from the common denominator approach is:
\[
1 - x^3 + x = 5 - x^3
\]
Among the given choices, the correct one representing the equivalent expression after multiplying by the common denominator \( x^2 \) is:
\[
x^2\left(\frac{1}{x^2} - x\right) + x^2\left(\frac{1}{x}\right) = x^2\left(\frac{5}{x^2} - x\right)
\]
Thus, the closest response would be:
\[
x^2\left( \frac{1}{x^2} - x \right) + x^2\left( \frac{1}{x} \right) = x^2\left( \frac{5}{x^2} - x \right)
\]