Solve 1/2x^3=6-3x^3

$$\frac { 1 } { 2 x ^ { 3 } } = 6 - 3 x ^ { 3 }$$

Solve for x

$$x=\frac{6^{\frac{2}{3}}\sqrt[3]{6-\sqrt{30}}}{6}\approx 0.44332378$$

$$x = \frac{6 ^ {\frac{2}{3}} \sqrt[3]{\sqrt{30} + 6}}{6} \approx 1.241352783$$

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 2x^{3}.

$$1=2x^{3}\times 6-3x^{3}\times 2x^{3}$$

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

$$1=2x^{3}\times 6-3x^{6}\times 2$$

Multiply 2 and 6 to get 12.

$$1=12x^{3}-3x^{6}\times 2$$

Multiply -3 and 2 to get -6.

$$1=12x^{3}-6x^{6}$$

Swap sides so that all variable terms are on the left hand side.

$$12x^{3}-6x^{6}=1$$

Subtract 1 from both sides.

$$12x^{3}-6x^{6}-1=0$$

Substitute t for x^{3}.

$$-6t^{2}+12t-1=0$$

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 -6 for a, 12 for b, and -1 for c in the quadratic formula.

$$t=\frac{-12±\sqrt{12^{2}-4\left(-6\right)\left(-1\right)}}{-6\times 2}$$

Do the calculations.

$$t=\frac{-12±2\sqrt{30}}{-12}$$

Solve the equation t=\frac{-12±2\sqrt{30}}{-12} when ± is plus and when ± is minus.

$$t=-\frac{\sqrt{30}}{6}+1 t=\frac{\sqrt{30}}{6}+1$$

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

$$x=\sqrt[3]{-\frac{\sqrt{30}}{6}+1} x=\sqrt[3]{\frac{\sqrt{30}}{6}+1}$$