Solve 3x^4+47x^2-14=0

$$3 x ^ { 4 } + 47 x ^ { 2 } - 14 = 0$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{6\left(\sqrt{2377}+47\right)}}{6}\approx 3.994881871i$$

$$x=-\frac{i\sqrt{6\left(\sqrt{2377}+47\right)}}{6}\approx -0-3.994881871i$$

$$x=-\frac{\sqrt{6\left(\sqrt{2377}-47\right)}}{6}\approx -0.540753637$$

$$x=\frac{\sqrt{6\left(\sqrt{2377}-47\right)}}{6}\approx 0.540753637$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}+47t-14=0$$

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 3 for a, 47 for b, and -14 for c in the quadratic formula.

$$t=\frac{-47±\sqrt{47^{2}-4\times 3\left(-14\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-47±\sqrt{2377}}{6}$$

Solve the equation t=\frac{-47±\sqrt{2377}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{2377}-47}{6} t=\frac{-\sqrt{2377}-47}{6}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=-\sqrt{\frac{\sqrt{2377}-47}{6}} x=\sqrt{\frac{\sqrt{2377}-47}{6}} x=-i\sqrt{\frac{\sqrt{2377}+47}{6}} x=i\sqrt{\frac{\sqrt{2377}+47}{6}}$$

Solve for x

$$x=-\frac{\sqrt{6\left(\sqrt{2377}-47\right)}}{6}\approx -0.540753637$$

$$x=\frac{\sqrt{6\left(\sqrt{2377}-47\right)}}{6}\approx 0.540753637$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}+47t-14=0$$

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 3 for a, 47 for b, and -14 for c in the quadratic formula.

$$t=\frac{-47±\sqrt{47^{2}-4\times 3\left(-14\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-47±\sqrt{2377}}{6}$$

Solve the equation t=\frac{-47±\sqrt{2377}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{2377}-47}{6} t=\frac{-\sqrt{2377}-47}{6}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=\frac{\sqrt{\frac{2\sqrt{2377}-94}{3}}}{2} x=-\frac{\sqrt{\frac{2\sqrt{2377}-94}{3}}}{2}$$