Solving Equations with Fractions: A Step-by-Step Guide

Equation: x^2 - frac{1}{x} frac{2x}{x^2 1} 3 Simplified Form: x^2 - frac{1}{x} frac{2x}{x^2 1} 3

The equation x^2 - frac{1}{x} frac{2x}{x^2 1} 3 is a complex rational equation. To solve it, we first need to simplify and transform it into a more manageable polynomial form. This involves finding a common denominator and then simplifying the equation.

Step 1: Eliminating the Denominators

To eliminate the denominators, we multiply both sides of the equation by the common denominator, which is (x^2 1)x. This gives us:

(x^2 1)x left( x^2 - frac{1}{x} frac{2x}{x^2 1} right) 3(x^2 1)x

After distributing, we simplify the left side:

x^4 - x 2x 3x^3 3x

Combining like terms, we get:

x^4 - 3x^3 - x^2 - 3x - 1 0

Step 2: Factoring the Polynomial

We now have the polynomial equation:

x^4 - 3x^3 - x^2 - 3x - 1 0

To solve this polynomial, we can use various methods. One approach is to factor the polynomial or use the Rational Root Theorem to find potential roots.

Step 3: Finding the Roots

The equation can be further simplified and solved by using the Rational Root Theorem, which suggests that any rational root of the polynomial is a factor of the constant term (in this case, -1) divided by a factor of the leading coefficient (which is 1). Therefore, the possible rational roots are ±1.

Testing x 1, we get:

1^4 - 3(1)^3 - (1)^2 - 3(1) - 1 1 - 3 - 1 - 3 - 1 -7 ≠ 0

This is not a root. However, testing x -1, we get:

(-1)^4 - 3(-1)^3 - (-1)^2 - 3(-1) - 1 1 3 - 1 3 - 1 6 ≠ 0

Since neither of these is a root, we need to use other methods. We can reduce the polynomial by dividing it by a simpler form or use a substitution method.

Dividing the polynomial by x - 1, we get:

x^3 - 2x^2 - 2x - 1 0

Dividing again by x - 1, we get:

x^2 - x - 1 0

This is a quadratic equation, which we can solve using the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 1, b -1, and c -1. Substituting these values, we get:

x frac{1 pm sqrt{1 4}}{2} frac{1 pm sqrt{5}}{2}

This gives us two complex roots:

x frac{1 sqrt{5}}{2} and x frac{1 - sqrt{5}}{2}

Conclusion

The roots of the equation are:

x 1, x frac{1 sqrt{5}}{2}, and x frac{1 - sqrt{5}}{2}

Therefore, the solutions to the equation x^2 - frac{1}{x} frac{2x}{x^2 1} 3 are x 1, x frac{1 sqrt{5}}{2}, and x frac{1 - sqrt{5}}{2}.