Solve 7x^4-37x^2+12=0

$$7 x ^ { 4 } - 37 x ^ { 2 } + 12 = 0$$

Solve for x

$$x=-\frac{\sqrt{518-14\sqrt{1033}}}{14}\approx -0.589169066$$

$$x=\frac{\sqrt{518-14\sqrt{1033}}}{14}\approx 0.589169066$$

$$x = \frac{\sqrt{14 \sqrt{1033} + 518}}{14} \approx 2.222294782$$

$$x = -\frac{\sqrt{14 \sqrt{1033} + 518}}{14} \approx -2.222294782$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$7t^{2}-37t+12=0$$

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 7 for a, -37 for b, and 12 for c in the quadratic formula.

$$t=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 7\times 12}}{2\times 7}$$

Do the calculations.

$$t=\frac{37±\sqrt{1033}}{14}$$

Solve the equation t=\frac{37±\sqrt{1033}}{14} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{1033}+37}{14} t=\frac{37-\sqrt{1033}}{14}$$

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

$$x=\frac{\sqrt{\frac{2\sqrt{1033}+74}{7}}}{2} x=-\frac{\sqrt{\frac{2\sqrt{1033}+74}{7}}}{2} x=\frac{\sqrt{\frac{74-2\sqrt{1033}}{7}}}{2} x=-\frac{\sqrt{\frac{74-2\sqrt{1033}}{7}}}{2}$$