Question
Solve the rational equation for x: 3/x+x−3/x=1/x^2 .
(1 point)
Responses
x=±1
x is equal to plus or minus 1
x = 0
x = 0
x = 1
x = 1
x = 3
(1 point)
Responses
x=±1
x is equal to plus or minus 1
x = 0
x = 0
x = 1
x = 1
x = 3
Answers
GPT-4o mini
To solve the equation \( \frac{3}{x} + x - \frac{3}{x} = \frac{1}{x^2} \), we can first simplify it.
Notice that \( \frac{3}{x} - \frac{3}{x} = 0 \), so the equation reduces to:
\[
x = \frac{1}{x^2}.
\]
Now we can multiply both sides by \( x^2 \) (assuming \( x \neq 0 \)):
\[
x^3 = 1.
\]
This gives us:
\[
x^3 - 1 = 0.
\]
Factoring the equation, we have:
\[
(x - 1)(x^2 + x + 1) = 0.
\]
From this, we see that one solution is:
\[
x - 1 = 0 \implies x = 1.
\]
To find the other solutions, we solve \( x^2 + x + 1 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}.
\]
The solutions for the quadratic are complex and not real numbers.
The only real solution for the original equation is:
\[
\boxed{1}.
\]
Notice that \( \frac{3}{x} - \frac{3}{x} = 0 \), so the equation reduces to:
\[
x = \frac{1}{x^2}.
\]
Now we can multiply both sides by \( x^2 \) (assuming \( x \neq 0 \)):
\[
x^3 = 1.
\]
This gives us:
\[
x^3 - 1 = 0.
\]
Factoring the equation, we have:
\[
(x - 1)(x^2 + x + 1) = 0.
\]
From this, we see that one solution is:
\[
x - 1 = 0 \implies x = 1.
\]
To find the other solutions, we solve \( x^2 + x + 1 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}.
\]
The solutions for the quadratic are complex and not real numbers.
The only real solution for the original equation is:
\[
\boxed{1}.
\]