r^4+6r^2+1=0

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Solution for r^4+6r^2+1=0 equation:


Simplifying
r4 + 6r2 + 1 = 0

Reorder the terms:
1 + 6r2 + r4 = 0

Solving
1 + 6r2 + r4 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 6r2 + -1 + r4 = 0 + -1

Reorder the terms:
1 + -1 + 6r2 + r4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 6r2 + r4 = 0 + -1
6r2 + r4 = 0 + -1

Combine like terms: 0 + -1 = -1
6r2 + r4 = -1

The r term is 6r2.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6r2 + 9 + r4 = -1 + 9

Reorder the terms:
9 + 6r2 + r4 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + 6r2 + r4 = 8

Factor a perfect square on the left side:
(r2 + 3)(r2 + 3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(r2 + 3) equal to 2.828427125 and -2.828427125.

Subproblem 1

r2 + 3 = 2.828427125 Simplifying r2 + 3 = 2.828427125 Reorder the terms: 3 + r2 = 2.828427125 Solving 3 + r2 = 2.828427125 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r2 = 2.828427125 + -3 Combine like terms: 3 + -3 = 0 0 + r2 = 2.828427125 + -3 r2 = 2.828427125 + -3 Combine like terms: 2.828427125 + -3 = -0.171572875 r2 = -0.171572875 Simplifying r2 = -0.171572875 Reorder the terms: 0.171572875 + r2 = -0.171572875 + 0.171572875 Combine like terms: -0.171572875 + 0.171572875 = 0.000000000 0.171572875 + r2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined.

Subproblem 2

r2 + 3 = -2.828427125 Simplifying r2 + 3 = -2.828427125 Reorder the terms: 3 + r2 = -2.828427125 Solving 3 + r2 = -2.828427125 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r2 = -2.828427125 + -3 Combine like terms: 3 + -3 = 0 0 + r2 = -2.828427125 + -3 r2 = -2.828427125 + -3 Combine like terms: -2.828427125 + -3 = -5.828427125 r2 = -5.828427125 Simplifying r2 = -5.828427125 Reorder the terms: 5.828427125 + r2 = -5.828427125 + 5.828427125 Combine like terms: -5.828427125 + 5.828427125 = 0.000000000 5.828427125 + r2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.

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