Solve 6x^4+29x^2+29=0

$$6 x ^ { 4 } + 29 x ^ { 2 } + 29 = 0$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{3\sqrt{145}+87}}{6}\approx 1.849360128i$$

$$x=-\frac{i\sqrt{3\sqrt{145}+87}}{6}\approx -0-1.849360128i$$

$$x=-\frac{i\sqrt{87-3\sqrt{145}}}{6}\approx -0-1.188781078i$$

$$x=\frac{i\sqrt{87-3\sqrt{145}}}{6}\approx 1.188781078i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$6t^{2}+29t+29=0$$

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 6 for a, 29 for b, and 29 for c in the quadratic formula.

$$t=\frac{-29±\sqrt{29^{2}-4\times 6\times 29}}{2\times 6}$$

Do the calculations.

$$t=\frac{-29±\sqrt{145}}{12}$$

Solve the equation t=\frac{-29±\sqrt{145}}{12} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{145}-29}{12} t=\frac{-\sqrt{145}-29}{12}$$

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

$$x=-i\sqrt{\frac{29-\sqrt{145}}{12}} x=i\sqrt{\frac{29-\sqrt{145}}{12}} x=-i\sqrt{\frac{\sqrt{145}+29}{12}} x=i\sqrt{\frac{\sqrt{145}+29}{12}}$$