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

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

Solve for x (complex solution)

$$x\in \frac{2^{\frac{2}{3}}\sqrt[3]{\sqrt{5}+3}e^{\frac{2\pi i}{3}}}{2},\frac{2^{\frac{2}{3}}\sqrt[3]{\sqrt{5}+3}e^{\frac{4\pi i}{3}}}{2},\frac{2^{\frac{2}{3}}\sqrt[3]{\sqrt{5}+3}}{2},\frac{2^{\frac{2}{3}}\sqrt[3]{3-\sqrt{5}}e^{\frac{4\pi i}{3}}}{2},\frac{2^{\frac{2}{3}}\sqrt[3]{3-\sqrt{5}}e^{\frac{2\pi i}{3}}}{2},\frac{2^{\frac{2}{3}}\sqrt[3]{3-\sqrt{5}}}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{3}.

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

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$x^{6}+1=3x^{3}$$

Subtract 3x^{3} from both sides.

$$x^{6}+1-3x^{3}=0$$

Substitute t for x^{3}.

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

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, -3 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{3±\sqrt{5}}{2}$$

Solve the equation t=\frac{3±\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{5}+3}{2} t=\frac{3-\sqrt{5}}{2}$$

Since x=t^{3}, the solutions are obtained by solving the equation for each t.

$$x=-\sqrt[3]{\frac{\sqrt{5}+3}{2}}e^{\frac{\pi i}{3}} x=\sqrt[3]{\frac{\sqrt{5}+3}{2}}ie^{\frac{\pi i}{6}} x=\sqrt[3]{\frac{\sqrt{5}+3}{2}} x=-\sqrt[3]{\frac{3-\sqrt{5}}{2}}e^{\frac{\pi i}{3}} x=\sqrt[3]{\frac{3-\sqrt{5}}{2}}ie^{\frac{\pi i}{6}} x=\sqrt[3]{\frac{3-\sqrt{5}}{2}}$$

Variable x cannot be equal to 0.

$$x=\sqrt[3]{\frac{3-\sqrt{5}}{2}} x=\sqrt[3]{\frac{3-\sqrt{5}}{2}}ie^{\frac{\pi i}{6}}\text{, }x\neq 0 x=-\sqrt[3]{\frac{3-\sqrt{5}}{2}}e^{\frac{\pi i}{3}}\text{, }x\neq 0 x=\sqrt[3]{\frac{\sqrt{5}+3}{2}} x=\sqrt[3]{\frac{\sqrt{5}+3}{2}}ie^{\frac{\pi i}{6}}\text{, }x\neq 0 x=-\sqrt[3]{\frac{\sqrt{5}+3}{2}}e^{\frac{\pi i}{3}}\text{, }x\neq 0$$

Solve for x

$$x=\frac{2^{\frac{2}{3}}\sqrt[3]{3-\sqrt{5}}}{2}\approx 0.72556263$$

$$x = \frac{2 ^ {\frac{2}{3}} \sqrt[3]{\sqrt{5} + 3}}{2} \approx 1.378240772$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{3}.

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

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$x^{6}+1=3x^{3}$$

Subtract 3x^{3} from both sides.

$$x^{6}+1-3x^{3}=0$$

Substitute t for x^{3}.

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

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, -3 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{3±\sqrt{5}}{2}$$

Solve the equation t=\frac{3±\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{5}+3}{2} t=\frac{3-\sqrt{5}}{2}$$

Since x=t^{3}, the solutions are obtained by evaluating x=\sqrt[3]{t} for each t.

$$x=\sqrt[3]{\frac{\sqrt{5}+3}{2}} x=\sqrt[3]{\frac{3-\sqrt{5}}{2}}$$