Solve x^4-6x^2+7=0

$$x ^ { 4 } - 6 x ^ { 2 } + 7 = 0$$

Solve for x

$$x=-\sqrt{3-\sqrt{2}}\approx -1.259280127$$

$$x=\sqrt{3-\sqrt{2}}\approx 1.259280127$$

$$x=\sqrt{\sqrt{2}+3}\approx 2.10100299$$

$$x=-\sqrt{\sqrt{2}+3}\approx -2.10100299$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-6t+7=0$$

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, -6 for b, and 7 for c in the quadratic formula.

$$t=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 1\times 7}}{2}$$

Do the calculations.

$$t=\frac{6±2\sqrt{2}}{2}$$

Solve the equation t=\frac{6±2\sqrt{2}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{2}+3 t=3-\sqrt{2}$$

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

$$x=\sqrt{\sqrt{2}+3} x=-\sqrt{\sqrt{2}+3} x=\sqrt{3-\sqrt{2}} x=-\sqrt{3-\sqrt{2}}$$