Solve x^2-5x^4/2-3=0

$$x ^ { 2 } - \frac { 5 x ^ { 4 } } { 2 } - 3 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt[4]{750}e^{\frac{-\arctan(\sqrt{29})i+2\pi i}{2}}}{5}\approx -0.804812126+0.669120735i$$

$$x=\frac{\sqrt[4]{750}e^{-\frac{\arctan(\sqrt{29})i}{2}}}{5}\approx 0.804812126-0.669120735i$$

$$x=\frac{\sqrt[4]{750}e^{\frac{\arctan(\sqrt{29})i+2\pi i}{2}}}{5}\approx -0.804812126-0.669120735i$$

$$x=\frac{\sqrt[4]{750}e^{\frac{\arctan(\sqrt{29})i}{2}}}{5}\approx 0.804812126+0.669120735i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2x^{2}-5x^{4}-6=0$$

Substitute t for x^{2}.

$$-5t^{2}+2t-6=0$$

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 -5 for a, 2 for b, and -6 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\left(-5\right)\left(-6\right)}}{-5\times 2}$$

Do the calculations.

$$t=\frac{-2±\sqrt{-116}}{-10}$$

Solve the equation t=\frac{-2±\sqrt{-116}}{-10} when ± is plus and when ± is minus.

$$t=\frac{-\sqrt{29}i+1}{5} t=\frac{1+\sqrt{29}i}{5}$$

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

$$x=\frac{\sqrt[4]{750}e^{-\frac{\arctan(\sqrt{29})i}{2}}}{5} x=\frac{\sqrt[4]{750}e^{\frac{-\arctan(\sqrt{29})i+2\pi i}{2}}}{5} x=\frac{\sqrt[4]{750}e^{\frac{\arctan(\sqrt{29})i+2\pi i}{2}}}{5} x=\frac{\sqrt[4]{750}e^{\frac{\arctan(\sqrt{29})i}{2}}}{5}$$