Solve x^2=2/x+1

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

Solve for x (complex solution)

$$x=-1-i$$

$$x=1$$

$$x=-1+i$$

Solution Steps

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$\left(x+1\right)x^{2}=2$$

Use the distributive property to multiply x+1 by x^{2}.

$$x^{3}+x^{2}=2$$

Subtract 2 from both sides.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±2,±1$$

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^{3}+x^{2}-2 by x-1 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.

$$x=\frac{-2±\sqrt{2^{2}-4\times 1\times 2}}{2}$$

Do the calculations.

$$x=\frac{-2±\sqrt{-4}}{2}$$

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

$$x=-1-i x=-1+i$$

List all found solutions.

$$x=1 x=-1-i x=-1+i$$

Solve for x

$$x=1$$

Solution Steps

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$\left(x+1\right)x^{2}=2$$

Use the distributive property to multiply x+1 by x^{2}.

$$x^{3}+x^{2}=2$$

Subtract 2 from both sides.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±2,±1$$

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^{3}+x^{2}-2 by x-1 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.

$$x=\frac{-2±\sqrt{2^{2}-4\times 1\times 2}}{2}$$

Do the calculations.

$$x=\frac{-2±\sqrt{-4}}{2}$$

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.

$$x=1$$