Solve x^4-20x^2-19=0

$${ x }^{ 4 } -20 { x }^{ 2 } -19=0$$

Solve for x (complex solution)

$$x=i\sqrt{\sqrt{119}-10}\approx 0.953263927i$$

$$x=-i\sqrt{\sqrt{119}-10}\approx -0-0.953263927i$$

$$x=-\sqrt{\sqrt{119}+10}\approx -4.572604522$$

$$x=\sqrt{\sqrt{119}+10}\approx 4.572604522$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-20t-19=0$$

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, -20 for b, and -19 for c in the quadratic formula.

$$t=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 1\left(-19\right)}}{2}$$

Do the calculations.

$$t=\frac{20±2\sqrt{119}}{2}$$

Solve the equation t=\frac{20±2\sqrt{119}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{119}+10 t=10-\sqrt{119}$$

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

$$x=-\sqrt{\sqrt{119}+10} x=\sqrt{\sqrt{119}+10} x=-i\sqrt{-\left(10-\sqrt{119}\right)} x=i\sqrt{-\left(10-\sqrt{119}\right)}$$

Solve for x

$$x=-\sqrt{\sqrt{119}+10}\approx -4.572604522$$

$$x=\sqrt{\sqrt{119}+10}\approx 4.572604522$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-20t-19=0$$

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, -20 for b, and -19 for c in the quadratic formula.

$$t=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 1\left(-19\right)}}{2}$$

Do the calculations.

$$t=\frac{20±2\sqrt{119}}{2}$$

Solve the equation t=\frac{20±2\sqrt{119}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{119}+10 t=10-\sqrt{119}$$

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

$$x=\sqrt{\sqrt{119}+10} x=-\sqrt{\sqrt{119}+10}$$