Solve a^2+a^4=4

$$a ^ { 2 } + a ^ { 4 } = 4$$

Solve for a

$$a = -\frac{\sqrt{2 {(\sqrt{17} - 1)}}}{2} \approx -1.249621068$$

$$a = \frac{\sqrt{2 {(\sqrt{17} - 1)}}}{2} \approx 1.249621068$$

Steps Using the Quadratic Formula

Subtract 4 from both sides.

$$a^{2}+a^{4}-4=0$$

Substitute t for a^{2}.

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

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

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

Do the calculations.

$$t=\frac{-1±\sqrt{17}}{2}$$

Solve the equation t=\frac{-1±\sqrt{17}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{17}-1}{2} t=\frac{-\sqrt{17}-1}{2}$$

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

$$a=\frac{\sqrt{2\sqrt{17}-2}}{2} a=-\frac{\sqrt{2\sqrt{17}-2}}{2}$$