Solve x^4-400x^2+10000=0

$$x ^ { 4 } - 400 x ^ { 2 } + 10000 = 0$$

Solve for x

$$x=5\left(\sqrt{2}-\sqrt{6}\right)\approx -5.176380902$$

$$x=5\left(\sqrt{6}-\sqrt{2}\right)\approx 5.176380902$$

$$x=5\left(\sqrt{2}+\sqrt{6}\right)\approx 19.318516526$$

$$x=5\left(-\sqrt{2}-\sqrt{6}\right)\approx -19.318516526$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-400t+10000=0$$

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, -400 for b, and 10000 for c in the quadratic formula.

$$t=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}-4\times 1\times 10000}}{2}$$

Do the calculations.

$$t=\frac{400±200\sqrt{3}}{2}$$

Solve the equation t=\frac{400±200\sqrt{3}}{2} when ± is plus and when ± is minus.

$$t=100\sqrt{3}+200 t=200-100\sqrt{3}$$

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

$$x=5\sqrt{2}+5\sqrt{6} x=-\left(5\sqrt{2}+5\sqrt{6}\right) x=-\left(5\sqrt{2}-5\sqrt{6}\right) x=5\sqrt{2}-5\sqrt{6}$$