Solve 8(x^3-1/x^3)=63

$$8 ( x ^ { 3 } - \frac { 1 } { x ^ { 3 } } ) = 63$$

Solve for x (complex solution)

$$x\in \frac{1+\sqrt{3}i}{4},-\frac{1}{2},\frac{-\sqrt{3}i+1}{4},-1+\sqrt{3}i,-\sqrt{3}i-1,2$$

Steps Using the Quadratic Formula

Divide both sides by 8.

$$x^{3}-\frac{1}{x^{3}}=\frac{63}{8}$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8x^{3}, the least common multiple of x^{3},8.

$$8x^{3}x^{3}-8=63x^{3}$$

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

$$8x^{6}-8=63x^{3}$$

Subtract 63x^{3} from both sides.

$$8x^{6}-8-63x^{3}=0$$

Substitute t for x^{3}.

$$8t^{2}-63t-8=0$$

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 8 for a, -63 for b, and -8 for c in the quadratic formula.

$$t=\frac{-\left(-63\right)±\sqrt{\left(-63\right)^{2}-4\times 8\left(-8\right)}}{2\times 8}$$

Do the calculations.

$$t=\frac{63±65}{16}$$

Solve the equation t=\frac{63±65}{16} when ± is plus and when ± is minus.

$$t=8 t=-\frac{1}{8}$$

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

$$x=-1+\sqrt{3}i x=-\sqrt{3}i-1 x=2 x=-\frac{1}{2} x=\frac{1+\sqrt{3}i}{4} x=\frac{-\sqrt{3}i+1}{4}$$

Solve for x

$$x=2$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using the Quadratic Formula

Divide both sides by 8.

$$x^{3}-\frac{1}{x^{3}}=\frac{63}{8}$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8x^{3}, the least common multiple of x^{3},8.

$$8x^{3}x^{3}-8=63x^{3}$$

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

$$8x^{6}-8=63x^{3}$$

Subtract 63x^{3} from both sides.

$$8x^{6}-8-63x^{3}=0$$

Substitute t for x^{3}.

$$8t^{2}-63t-8=0$$

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 8 for a, -63 for b, and -8 for c in the quadratic formula.

$$t=\frac{-\left(-63\right)±\sqrt{\left(-63\right)^{2}-4\times 8\left(-8\right)}}{2\times 8}$$

Do the calculations.

$$t=\frac{63±65}{16}$$

Solve the equation t=\frac{63±65}{16} when ± is plus and when ± is minus.

$$t=8 t=-\frac{1}{8}$$

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

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