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

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

Solve for x

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

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

$$x=-1$$

Solution Steps

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

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

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

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

Add x^{2} to both sides.

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

Combine 2x^{2} and x^{2} to get 3x^{2}.

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

Subtract 2 from both sides.

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

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}+3x^{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.

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

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.

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

List all found solutions.

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