Solve x=2x+1/x^2-2x+4

$$x = \frac { 2 x + 1 } { x ^ { 2 } - 2 x + 4 }$$

Solve for x

$$x=1$$

Solution Steps

Subtract \frac{2x+1}{x^{2}-2x+4} from both sides.

$$x-\frac{2x+1}{x^{2}-2x+4}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x^{2}-2x+4}{x^{2}-2x+4}.

$$\frac{x\left(x^{2}-2x+4\right)}{x^{2}-2x+4}-\frac{2x+1}{x^{2}-2x+4}=0$$

Since \frac{x\left(x^{2}-2x+4\right)}{x^{2}-2x+4} and \frac{2x+1}{x^{2}-2x+4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x^{2}-2x+4\right)-\left(2x+1\right)}{x^{2}-2x+4}=0$$

Do the multiplications in x\left(x^{2}-2x+4\right)-\left(2x+1\right).

$$\frac{x^{3}-2x^{2}+4x-2x-1}{x^{2}-2x+4}=0$$

Combine like terms in x^{3}-2x^{2}+4x-2x-1.

$$\frac{x^{3}-2x^{2}+2x-1}{x^{2}-2x+4}=0$$

Multiply both sides of the equation by x^{2}-2x+4.

$$x^{3}-2x^{2}+2x-1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-2x^{2}+2x-1 by x-1 to get x^{2}-x+1. Solve the equation where the result equals to 0.

$$x^{2}-x+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -1 for b, and 1 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$x=\frac{1±\sqrt{-3}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=1$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{3}i+1}{2}\approx 0.5-0.866025404i$$

$$x=1$$

$$x=\frac{1+\sqrt{3}i}{2}\approx 0.5+0.866025404i$$