x^4-65x^2+16=0

Solution for x^4-65x^2+16=0 equation:

Simplifying
x4 + -65x2 + 16 = 0

Reorder the terms:
16 + -65x2 + x4 = 0

Solving
16 + -65x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + -65x2 + -16 + x4 = 0 + -16

Reorder the terms:
16 + -16 + -65x2 + x4 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + -65x2 + x4 = 0 + -16
-65x2 + x4 = 0 + -16

Combine like terms: 0 + -16 = -16
-65x2 + x4 = -16

The x term is -65x2.  Take half its coefficient (-32.5).
Square it (1056.25) and add it to both sides.

Add '1056.25' to each side of the equation.
-65x2 + 1056.25 + x4 = -16 + 1056.25

Reorder the terms:
1056.25 + -65x2 + x4 = -16 + 1056.25

Combine like terms: -16 + 1056.25 = 1040.25
1056.25 + -65x2 + x4 = 1040.25

Factor a perfect square on the left side:
(x2 + -32.5)(x2 + -32.5) = 1040.25

Calculate the square root of the right side: 32.252906846

Break this problem into two subproblems by setting 
(x2 + -32.5) equal to 32.252906846 and -32.252906846.

Subproblem 1
x2 + -32.5 = 32.252906846

Simplifying
x2 + -32.5 = 32.252906846

Reorder the terms:
-32.5 + x2 = 32.252906846

Solving
-32.5 + x2 = 32.252906846

Solving for variable 'x'.

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

Add '32.5' to each side of the equation.
-32.5 + 32.5 + x2 = 32.252906846 + 32.5

Combine like terms: -32.5 + 32.5 = 0.0
0.0 + x2 = 32.252906846 + 32.5
x2 = 32.252906846 + 32.5

Combine like terms: 32.252906846 + 32.5 = 64.752906846
x2 = 64.752906846

Simplifying
x2 = 64.752906846

Take the square root of each side:
x = {-8.04691909, 8.04691909}

Subproblem 2
x2 + -32.5 = -32.252906846

Simplifying
x2 + -32.5 = -32.252906846

Reorder the terms:
-32.5 + x2 = -32.252906846

Solving
-32.5 + x2 = -32.252906846

Solving for variable 'x'.

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

Add '32.5' to each side of the equation.
-32.5 + 32.5 + x2 = -32.252906846 + 32.5

Combine like terms: -32.5 + 32.5 = 0.0
0.0 + x2 = -32.252906846 + 32.5
x2 = -32.252906846 + 32.5

Combine like terms: -32.252906846 + 32.5 = 0.247093154
x2 = 0.247093154

Simplifying
x2 = 0.247093154

Take the square root of each side:
x = {-0.497084655, 0.497084655}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-8.04691909, 8.04691909, -0.497084655, 0.497084655}