Solve 1-1x^4-5x^2+4=0

$$1 - 1 x ^ { 4 } - 5 x ^ { 2 } + 4 = 0$$

Solve for x (complex solution)

$$x=\frac{5^{\frac{3}{4}}i+\sqrt[4]{5}i}{2}\approx 2.419525153i$$

$$x=\frac{-5^{\frac{3}{4}}i-\sqrt[4]{5}i}{2}\approx -2.419525153i$$

$$x=-\frac{\sqrt{6\sqrt{5}-10}}{2}\approx -0.924176372$$

$$x=\frac{\sqrt{6\sqrt{5}-10}}{2}\approx 0.924176372$$

Steps Using the Quadratic Formula

Add 1 and 4 to get 5.

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

Substitute t for x^{2}.

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

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

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 5}}{-2}$$

Do the calculations.

$$t=\frac{5±3\sqrt{5}}{-2}$$

Solve the equation t=\frac{5±3\sqrt{5}}{-2} when ± is plus and when ± is minus.

$$t=\frac{-3\sqrt{5}-5}{2} t=\frac{3\sqrt{5}-5}{2}$$

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

$$x=-i\sqrt{\frac{3\sqrt{5}+5}{2}} x=i\sqrt{\frac{3\sqrt{5}+5}{2}} x=-\sqrt{\frac{3\sqrt{5}-5}{2}} x=\sqrt{\frac{3\sqrt{5}-5}{2}}$$

Solve for x

$$x=-\frac{\sqrt{6\sqrt{5}-10}}{2}\approx -0.924176372$$

$$x=\frac{\sqrt{6\sqrt{5}-10}}{2}\approx 0.924176372$$

Steps Using the Quadratic Formula

Add 1 and 4 to get 5.

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

Substitute t for x^{2}.

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

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

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 5}}{-2}$$

Do the calculations.

$$t=\frac{5±3\sqrt{5}}{-2}$$

Solve the equation t=\frac{5±3\sqrt{5}}{-2} when ± is plus and when ± is minus.

$$t=\frac{-3\sqrt{5}-5}{2} t=\frac{3\sqrt{5}-5}{2}$$

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

$$x=\frac{\sqrt{6\sqrt{5}-10}}{2} x=-\frac{\sqrt{6\sqrt{5}-10}}{2}$$