Solve x^2-2-2x/x*x^2-2-1x/x=0

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

Solve for x

$$x=\sqrt{3}+1\approx 2.732050808$$

$$x=1-\sqrt{3}\approx -0.732050808$$

$$x=-1$$

$$x=2$$

Solution Steps

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(x^{2}-2-2x\right)\times \frac{x^{2}-2-x}{x}=0$$

Express \left(x^{2}-2-2x\right)\times \frac{x^{2}-2-x}{x} as a single fraction.

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

Use the distributive property to multiply x^{2}-2-2x by x^{2}-2-x and combine like terms.

$$\frac{x^{4}-2x^{2}-3x^{3}+4+6x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$x^{4}-2x^{2}-3x^{3}+4+6x=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{4}-3x^{3}-2x^{2}+6x+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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^{4}-3x^{3}-2x^{2}+6x+4 by x+1 to get x^{3}-4x^{2}+2x+4. Solve the equation where the result equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-4x^{2}+2x+4 by x-2 to get x^{2}-2x-2. Solve the equation where the result equals to 0.

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

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, -2 for b, and -2 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{2±2\sqrt{3}}{2}$$

Solve the equation x^{2}-2x-2=0 when ± is plus and when ± is minus.

$$x=1-\sqrt{3} x=\sqrt{3}+1$$

List all found solutions.

$$x=-1 x=2 x=1-\sqrt{3} x=\sqrt{3}+1$$