Solve x^6-4x^3-3=0

$$x ^ { 6 } - 4 x ^ { 3 } - 3 = 0$$

Solve for x (complex solution)

$$x\in \sqrt[3]{\sqrt{7}-2}e^{\frac{\pi i}{3}},\sqrt[3]{\sqrt{7}-2}e^{\frac{5\pi i}{3}},-\sqrt[3]{\sqrt{7}-2},\sqrt[3]{\sqrt{7}+2}e^{\frac{4\pi i}{3}},\sqrt[3]{\sqrt{7}+2},\sqrt[3]{\sqrt{7}+2}e^{\frac{2\pi i}{3}}$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$t^{2}-4t-3=0$$

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, -4 for b, and -3 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{4±2\sqrt{7}}{2}$$

Solve the equation t=\frac{4±2\sqrt{7}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{7}+2 t=2-\sqrt{7}$$

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

$$x=-\sqrt[3]{\sqrt{7}+2}e^{\frac{\pi i}{3}} x=\sqrt[3]{\sqrt{7}+2}ie^{\frac{\pi i}{6}} x=\sqrt[3]{\sqrt{7}+2} x=-\sqrt[3]{-\left(2-\sqrt{7}\right)}ie^{\frac{\pi i}{6}} x=-\sqrt[3]{-\left(2-\sqrt{7}\right)} x=\sqrt[3]{-\left(2-\sqrt{7}\right)}e^{\frac{\pi i}{3}}$$

Solve for x

$$x=\sqrt[3]{\sqrt{7}+2}\approx 1.668599027$$

$$x=\sqrt[3]{2-\sqrt{7}}\approx -0.864347603$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$t^{2}-4t-3=0$$

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, -4 for b, and -3 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{4±2\sqrt{7}}{2}$$

Solve the equation t=\frac{4±2\sqrt{7}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{7}+2 t=2-\sqrt{7}$$

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

$$x=\sqrt[3]{\sqrt{7}+2} x=\sqrt[3]{2-\sqrt{7}}$$