(x²+1)/x-x/(x²+1)=3/2

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To solve the equation, we first find a common denominator to combine the fractions on the left side of the equation:

[(x²+1)(x) - x(x)] / (x)(x²+1) = 3/2

Simplify the numerator:

(x^3 + x - x^2) / (x)(x²+1) = 3/2

Combine like terms in the numerator:

(x^3 - x^2 + x) / (x)(x²+1) = 3/2

Now, multiply both sides of the equation by (x)(x²+1):

(x^3 - x^2 + x) = 3/2 * (x)(x²+1)

Expand the right side:

x^3 - x^2 + x = (3/2)x^3 + (3/2)x

Multiply everything by 2 to clear the fraction:

2x^3 - 2x^2 + 2x = 3x^3 + 3x

Now, combine like terms:

-2x^2 + 2x = x^3 + 3x

Rearrange the equation to set it equal to zero:

x^3 + 2x^2 - x - 2x = 0

Factor out an x:

x(x^2 + 2x - 1) = 0

Now, we need to solve for x. Either x = 0 or we need to solve the quadratic equation x^2 + 2x - 1 = 0 using the quadratic formula. We get:

x = (-2 ± √(2² - 41(-1))) / (2*1)
x = (-2 ± √(4 + 4))/2
x = (-2 ± √8) / 2
x = (-2 ± 2√2) / 2
x = -1 ± √2

So, the solutions to the equation are x = 0, x = -1 + √2, and x = -1 - √2.