Question
Solve the rational equation for x: 3/x+x−3/x=1/x2 (1 point) Responses x=0 x=0 x=3 x=3 x=1 x=1 x=±1
Answers
GPT-4o mini
To solve the rational equation
\[
\frac{3}{x} + x - \frac{3}{x} = \frac{1}{x^2}
\]
we can simplify the left side first. The terms \(\frac{3}{x}\) and \(-\frac{3}{x}\) cancel out:
\[
x = \frac{1}{x^2}
\]
Next, we can eliminate the fraction by multiplying both sides by \(x^2\) (assuming \(x \neq 0\)):
\[
x \cdot x^2 = 1
\]
This simplifies to:
\[
x^3 = 1
\]
To find the solutions, we take the cube root of both sides:
\[
x = 1
\]
Now, we should also consider the possibility that \(x = 0\) could lead to any extraneous solutions or restrictions. However, substituting \(x = 0\) into the original equation would make the rational expressions undefined because division by zero is not allowed.
Thus, the only valid solution we found is:
\[
\boxed{1}
\]
\[
\frac{3}{x} + x - \frac{3}{x} = \frac{1}{x^2}
\]
we can simplify the left side first. The terms \(\frac{3}{x}\) and \(-\frac{3}{x}\) cancel out:
\[
x = \frac{1}{x^2}
\]
Next, we can eliminate the fraction by multiplying both sides by \(x^2\) (assuming \(x \neq 0\)):
\[
x \cdot x^2 = 1
\]
This simplifies to:
\[
x^3 = 1
\]
To find the solutions, we take the cube root of both sides:
\[
x = 1
\]
Now, we should also consider the possibility that \(x = 0\) could lead to any extraneous solutions or restrictions. However, substituting \(x = 0\) into the original equation would make the rational expressions undefined because division by zero is not allowed.
Thus, the only valid solution we found is:
\[
\boxed{1}
\]