Solve A_8^4-4A_8^2=1

$$A _ { 8 } ^ { 4 } - 4 A _ { 8 } ^ { 2 } = 1$$

Solve for A_8

$$A_{8}=-\sqrt{\sqrt{5}+2}\approx -2.058171027$$

$$A_{8}=\sqrt{\sqrt{5}+2}\approx 2.058171027$$

Steps Using the Quadratic Formula

Subtract 1 from both sides.

$$A_{8}^{4}-4A_{8}^{2}-1=0$$

Substitute t for A_{8}^{2}.

$$t^{2}-4t-1=0$$

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, -4 for b, and -1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{4±2\sqrt{5}}{2}$$

Solve the equation t=\frac{4±2\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{5}+2 t=2-\sqrt{5}$$

Since A_{8}=t^{2}, the solutions are obtained by evaluating A_{8}=±\sqrt{t} for positive t.

$$A_{8}=\sqrt{\sqrt{5}+2} A_{8}=-\sqrt{\sqrt{5}+2}$$