Solve 4k^2+4k^4+1=810

$$4 k ^ { 2 } + 4 k ^ { 4 } + 1 = 810$$

Solve for k

$$k = -\frac{\sqrt{18 \sqrt{10} - 2}}{2} \approx -3.705435126$$

$$k = \frac{\sqrt{18 \sqrt{10} - 2}}{2} \approx 3.705435126$$

Steps Using the Quadratic Formula

Subtract 810 from both sides.

$$4k^{2}+4k^{4}+1-810=0$$

Subtract 810 from 1 to get -809.

$$4k^{2}+4k^{4}-809=0$$

Substitute t for k^{2}.

$$4t^{2}+4t-809=0$$

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 4 for a, 4 for b, and -809 for c in the quadratic formula.

$$t=\frac{-4±\sqrt{4^{2}-4\times 4\left(-809\right)}}{2\times 4}$$

Do the calculations.

$$t=\frac{-4±36\sqrt{10}}{8}$$

Solve the equation t=\frac{-4±36\sqrt{10}}{8} when ± is plus and when ± is minus.

$$t=\frac{9\sqrt{10}-1}{2} t=\frac{-9\sqrt{10}-1}{2}$$

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

$$k=\frac{\sqrt{18\sqrt{10}-2}}{2} k=-\frac{\sqrt{18\sqrt{10}-2}}{2}$$