41a^4-120a^2+80=

Solution for 41a^4-120a^2+80= equation:

Simplifying
41a4 + -120a2 + 80 = 0

Reorder the terms:
80 + -120a2 + 41a4 = 0

Solving
80 + -120a2 + 41a4 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
41 the coefficient of the squared term: 

Divide each side by '41'.
1.951219512 + -2.926829268a2 + a4 = 0

Move the constant term to the right:

Add '-1.951219512' to each side of the equation.
1.951219512 + -2.926829268a2 + -1.951219512 + a4 = 0 + -1.951219512

Reorder the terms:
1.951219512 + -1.951219512 + -2.926829268a2 + a4 = 0 + -1.951219512

Combine like terms: 1.951219512 + -1.951219512 = 0.000000000
0.000000000 + -2.926829268a2 + a4 = 0 + -1.951219512
-2.926829268a2 + a4 = 0 + -1.951219512

Combine like terms: 0 + -1.951219512 = -1.951219512
-2.926829268a2 + a4 = -1.951219512

The a term is -2.926829268a2.  Take half its coefficient (-1.463414634).
Square it (2.141582391) and add it to both sides.

Add '2.141582391' to each side of the equation.
-2.926829268a2 + 2.141582391 + a4 = -1.951219512 + 2.141582391

Reorder the terms:
2.141582391 + -2.926829268a2 + a4 = -1.951219512 + 2.141582391

Combine like terms: -1.951219512 + 2.141582391 = 0.190362879
2.141582391 + -2.926829268a2 + a4 = 0.190362879

Factor a perfect square on the left side:
(a2 + -1.463414634)(a2 + -1.463414634) = 0.190362879

Calculate the square root of the right side: 0.436305947

Break this problem into two subproblems by setting 
(a2 + -1.463414634) equal to 0.436305947 and -0.436305947.

Subproblem 1
a2 + -1.463414634 = 0.436305947

Simplifying
a2 + -1.463414634 = 0.436305947

Reorder the terms:
-1.463414634 + a2 = 0.436305947

Solving
-1.463414634 + a2 = 0.436305947

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '1.463414634' to each side of the equation.
-1.463414634 + 1.463414634 + a2 = 0.436305947 + 1.463414634

Combine like terms: -1.463414634 + 1.463414634 = 0.000000000
0.000000000 + a2 = 0.436305947 + 1.463414634
a2 = 0.436305947 + 1.463414634

Combine like terms: 0.436305947 + 1.463414634 = 1.899720581
a2 = 1.899720581

Simplifying
a2 = 1.899720581

Take the square root of each side:
a = {-1.378303516, 1.378303516}

Subproblem 2
a2 + -1.463414634 = -0.436305947

Simplifying
a2 + -1.463414634 = -0.436305947

Reorder the terms:
-1.463414634 + a2 = -0.436305947

Solving
-1.463414634 + a2 = -0.436305947

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '1.463414634' to each side of the equation.
-1.463414634 + 1.463414634 + a2 = -0.436305947 + 1.463414634

Combine like terms: -1.463414634 + 1.463414634 = 0.000000000
0.000000000 + a2 = -0.436305947 + 1.463414634
a2 = -0.436305947 + 1.463414634

Combine like terms: -0.436305947 + 1.463414634 = 1.027108687
a2 = 1.027108687

Simplifying
a2 = 1.027108687

Take the square root of each side:
a = {-1.013463708, 1.013463708}

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-1.378303516, 1.378303516, -1.013463708, 1.013463708}