Solve left(x^2right)^2-16x^2-256=0

$${ \left( { x }^{ 2 } \right) }^{ 2 } -16 { x }^{ 2 } -256=0$$

Solve for x (complex solution)

$$x=2i\sqrt{2\left(\sqrt{5}-1\right)}\approx 3.144605511i$$

$$x=-2i\sqrt{2\left(\sqrt{5}-1\right)}\approx -0-3.144605511i$$

$$x=-2\sqrt{2\left(\sqrt{5}+1\right)}\approx -5.088078598$$

$$x=2\sqrt{2\left(\sqrt{5}+1\right)}\approx 5.088078598$$

Steps Using the Quadratic Formula

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-16x^{2}-256=0$$

Substitute t for x^{2}.

$$t^{2}-16t-256=0$$

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, -16 for b, and -256 for c in the quadratic formula.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 1\left(-256\right)}}{2}$$

Do the calculations.

$$t=\frac{16±16\sqrt{5}}{2}$$

Solve the equation t=\frac{16±16\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=8\sqrt{5}+8 t=8-8\sqrt{5}$$

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

$$x=-\sqrt{8\sqrt{5}+8} x=\sqrt{8\sqrt{5}+8} x=-i\sqrt{-\left(8-8\sqrt{5}\right)} x=i\sqrt{-\left(8-8\sqrt{5}\right)}$$

Solve for x

$$x=-2\sqrt{2\left(\sqrt{5}+1\right)}\approx -5.088078598$$

$$x=2\sqrt{2\left(\sqrt{5}+1\right)}\approx 5.088078598$$

Steps Using the Quadratic Formula

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-16x^{2}-256=0$$

Substitute t for x^{2}.

$$t^{2}-16t-256=0$$

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, -16 for b, and -256 for c in the quadratic formula.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 1\left(-256\right)}}{2}$$

Do the calculations.

$$t=\frac{16±16\sqrt{5}}{2}$$

Solve the equation t=\frac{16±16\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=8\sqrt{5}+8 t=8-8\sqrt{5}$$

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

$$x=\frac{\sqrt{32\sqrt{5}+32}}{2} x=-\frac{\sqrt{32\sqrt{5}+32}}{2}$$