Solve x^4-73x^2+40=0

$$x ^ { 4 } - 73 x ^ { 2 } + 40 = 0$$

Solve for x

$$x=-\frac{\sqrt{146-2\sqrt{5169}}}{2}\approx -0.743048494$$

$$x=\frac{\sqrt{146-2\sqrt{5169}}}{2}\approx 0.743048494$$

$$x = \frac{\sqrt{2 \sqrt{5169} + 146}}{2} \approx 8.511631978$$

$$x = -\frac{\sqrt{2 \sqrt{5169} + 146}}{2} \approx -8.511631978$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-73t+40=0$$

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, -73 for b, and 40 for c in the quadratic formula.

$$t=\frac{-\left(-73\right)±\sqrt{\left(-73\right)^{2}-4\times 1\times 40}}{2}$$

Do the calculations.

$$t=\frac{73±\sqrt{5169}}{2}$$

Solve the equation t=\frac{73±\sqrt{5169}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{5169}+73}{2} t=\frac{73-\sqrt{5169}}{2}$$

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

$$x=\frac{\sqrt{2\sqrt{5169}+146}}{2} x=-\frac{\sqrt{2\sqrt{5169}+146}}{2} x=\frac{\sqrt{146-2\sqrt{5169}}}{2} x=-\frac{\sqrt{146-2\sqrt{5169}}}{2}$$