Solve 2x^6+3x^3-35=0

$$2 x ^ { 6 } + 3 x ^ { 3 } - 35 = 0$$

Solve for x (complex solution)

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

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$2t^{2}+3t-35=0$$

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 2 for a, 3 for b, and -35 for c in the quadratic formula.

$$t=\frac{-3±\sqrt{3^{2}-4\times 2\left(-35\right)}}{2\times 2}$$

Do the calculations.

$$t=\frac{-3±17}{4}$$

Solve the equation t=\frac{-3±17}{4} when ± is plus and when ± is minus.

$$t=\frac{7}{2} t=-5$$

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

$$x=-\frac{\sqrt[3]{7}\times 2^{\frac{2}{3}}e^{\frac{\pi i}{3}}}{2} x=\frac{\sqrt[3]{7}\times 2^{\frac{2}{3}}ie^{\frac{\pi i}{6}}}{2} x=\frac{\sqrt[3]{7}\times 2^{\frac{2}{3}}}{2} x=-\sqrt[3]{5}ie^{\frac{\pi i}{6}} x=-\sqrt[3]{5} x=\sqrt[3]{5}e^{\frac{\pi i}{3}}$$

Solve for x

$$x=-\sqrt[3]{5}\approx -1.709975947$$

$$x = \frac{\sqrt[3]{7} \cdot 2 ^ {\frac{2}{3}}}{2} \approx 1.518294486$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$2t^{2}+3t-35=0$$

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 2 for a, 3 for b, and -35 for c in the quadratic formula.

$$t=\frac{-3±\sqrt{3^{2}-4\times 2\left(-35\right)}}{2\times 2}$$

Do the calculations.

$$t=\frac{-3±17}{4}$$

Solve the equation t=\frac{-3±17}{4} when ± is plus and when ± is minus.

$$t=\frac{7}{2} t=-5$$

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

$$x=\frac{\sqrt[3]{7}\times 2^{\frac{2}{3}}}{2} x=-\sqrt[3]{5}$$