Solve x^2+x^2/1-x^3=1

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

Solve for x

$$x=\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

$$x=\frac{-\sqrt{5}-1}{2}\approx -1.618033989$$

$$x=-1$$

Solution Steps

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(-x^{2}-x-1\right).

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

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

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

Use the distributive property to multiply x-1 by -x^{2}-x-1 and combine like terms.

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

Add x^{3} to both sides.

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

Subtract 1 from both sides.

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

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

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

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}-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.

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

Do the calculations.

$$x=\frac{-1±\sqrt{5}}{2}$$

Solve the equation x^{2}+x-1=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$

List all found solutions.

$$x=-1 x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$