Solve x^4-4x^2-46=0

$$x ^ { 4 } - 4 x ^ { 2 } - 46 = 0$$

Solve for x (complex solution)

$$x=i\sqrt{5\sqrt{2}-2}\approx 2.251903153i$$

$$x=-i\sqrt{5\sqrt{2}-2}\approx -0-2.251903153i$$

$$x=-\sqrt{5\sqrt{2}+2}\approx -3.011821345$$

$$x=\sqrt{5\sqrt{2}+2}\approx 3.011821345$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-4t-46=0$$

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 -46 for c in the quadratic formula.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\left(-46\right)}}{2}$$

Do the calculations.

$$t=\frac{4±10\sqrt{2}}{2}$$

Solve the equation t=\frac{4±10\sqrt{2}}{2} when ± is plus and when ± is minus.

$$t=5\sqrt{2}+2 t=2-5\sqrt{2}$$

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

$$x=-\sqrt{5\sqrt{2}+2} x=\sqrt{5\sqrt{2}+2} x=-i\sqrt{-\left(2-5\sqrt{2}\right)} x=i\sqrt{-\left(2-5\sqrt{2}\right)}$$

Solve for x

$$x=-\sqrt{5\sqrt{2}+2}\approx -3.011821345$$

$$x=\sqrt{5\sqrt{2}+2}\approx 3.011821345$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-4t-46=0$$

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 -46 for c in the quadratic formula.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\left(-46\right)}}{2}$$

Do the calculations.

$$t=\frac{4±10\sqrt{2}}{2}$$

Solve the equation t=\frac{4±10\sqrt{2}}{2} when ± is plus and when ± is minus.

$$t=5\sqrt{2}+2 t=2-5\sqrt{2}$$

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

$$x=\sqrt{5\sqrt{2}+2} x=-\sqrt{5\sqrt{2}+2}$$