Solve 3x^4+47x^2-36=1

$$3 x ^ { 4 } + 47 x ^ { 2 } - 36 = 1$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{6\left(\sqrt{2653}+47\right)}}{6}\approx 4.051898342i$$

$$x=-\frac{i\sqrt{6\left(\sqrt{2653}+47\right)}}{6}\approx -0-4.051898342i$$

$$x=-\frac{\sqrt{6\left(\sqrt{2653}-47\right)}}{6}\approx -0.866725739$$

$$x=\frac{\sqrt{6\left(\sqrt{2653}-47\right)}}{6}\approx 0.866725739$$

Steps Using the Quadratic Formula

Subtract 1 from both sides.

$$3x^{4}+47x^{2}-36-1=0$$

Subtract 1 from -36 to get -37.

$$3x^{4}+47x^{2}-37=0$$

Substitute t for x^{2}.

$$3t^{2}+47t-37=0$$

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 -37 for c in the quadratic formula.

$$t=\frac{-47±\sqrt{47^{2}-4\times 3\left(-37\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-47±\sqrt{2653}}{6}$$

Solve the equation t=\frac{-47±\sqrt{2653}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{2653}-47}{6} t=\frac{-\sqrt{2653}-47}{6}$$

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

$$x=-\sqrt{\frac{\sqrt{2653}-47}{6}} x=\sqrt{\frac{\sqrt{2653}-47}{6}} x=-i\sqrt{\frac{\sqrt{2653}+47}{6}} x=i\sqrt{\frac{\sqrt{2653}+47}{6}}$$

Solve for x

$$x=-\frac{\sqrt{6\left(\sqrt{2653}-47\right)}}{6}\approx -0.866725739$$

$$x=\frac{\sqrt{6\left(\sqrt{2653}-47\right)}}{6}\approx 0.866725739$$

Steps Using the Quadratic Formula

Subtract 1 from both sides.

$$3x^{4}+47x^{2}-36-1=0$$

Subtract 1 from -36 to get -37.

$$3x^{4}+47x^{2}-37=0$$

Substitute t for x^{2}.

$$3t^{2}+47t-37=0$$

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 -37 for c in the quadratic formula.

$$t=\frac{-47±\sqrt{47^{2}-4\times 3\left(-37\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-47±\sqrt{2653}}{6}$$

Solve the equation t=\frac{-47±\sqrt{2653}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{2653}-47}{6} t=\frac{-\sqrt{2653}-47}{6}$$

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

$$x=\frac{\sqrt{\frac{2\sqrt{2653}-94}{3}}}{2} x=-\frac{\sqrt{\frac{2\sqrt{2653}-94}{3}}}{2}$$