Solve a^4-110a^2+900=0

$$a ^ { 4 } - 110 a ^ { 2 } + 900 = 0$$

Solve for a

$$a=\frac{5\sqrt{2}-\sqrt{170}}{2}\approx -2.983668499$$

$$a = \frac{\sqrt{170} - 5 \sqrt{2}}{2} \approx 2.983668499$$

$$a = \frac{\sqrt{170} + 5 \sqrt{2}}{2} \approx 10.054736311$$

$$a=\frac{-\sqrt{170}-5\sqrt{2}}{2}\approx -10.054736311$$

Steps Using the Quadratic Formula

Substitute t for a^{2}.

$$t^{2}-110t+900=0$$

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, -110 for b, and 900 for c in the quadratic formula.

$$t=\frac{-\left(-110\right)±\sqrt{\left(-110\right)^{2}-4\times 1\times 900}}{2}$$

Do the calculations.

$$t=\frac{110±10\sqrt{85}}{2}$$

Solve the equation t=\frac{110±10\sqrt{85}}{2} when ± is plus and when ± is minus.

$$t=5\sqrt{85}+55 t=55-5\sqrt{85}$$

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

$$a=\frac{\sqrt{170}+5\sqrt{2}}{2} a=-\frac{\sqrt{170}+5\sqrt{2}}{2} a=-\frac{5\sqrt{2}-\sqrt{170}}{2} a=\frac{5\sqrt{2}-\sqrt{170}}{2}$$