x^4-72x^2+96=0

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Solution for x^4-72x^2+96=0 equation: 


Simplifying
x4 + -72x2 + 96 = 0

Reorder the terms:
96 + -72x2 + x4 = 0

Solving
96 + -72x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-96' to each side of the equation.
96 + -72x2 + -96 + x4 = 0 + -96

Reorder the terms:
96 + -96 + -72x2 + x4 = 0 + -96

Combine like terms: 96 + -96 = 0
0 + -72x2 + x4 = 0 + -96
-72x2 + x4 = 0 + -96

Combine like terms: 0 + -96 = -96
-72x2 + x4 = -96

The x term is -72x2.  Take half its coefficient (-36).
Square it (1296) and add it to both sides.

Add '1296' to each side of the equation.
-72x2 + 1296 + x4 = -96 + 1296

Reorder the terms:
1296 + -72x2 + x4 = -96 + 1296

Combine like terms: -96 + 1296 = 1200
1296 + -72x2 + x4 = 1200

Factor a perfect square on the left side:
(x2 + -36)(x2 + -36) = 1200

Calculate the square root of the right side: 34.641016151

Break this problem into two subproblems by setting 
(x2 + -36) equal to 34.641016151 and -34.641016151.


Subproblem 1
x2 + -36 = 34.641016151

Simplifying
x2 + -36 = 34.641016151

Reorder the terms:
-36 + x2 = 34.641016151

Solving
-36 + x2 = 34.641016151

Solving for variable 'x'.

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

Add '36' to each side of the equation.
-36 + 36 + x2 = 34.641016151 + 36

Combine like terms: -36 + 36 = 0
0 + x2 = 34.641016151 + 36
x2 = 34.641016151 + 36

Combine like terms: 34.641016151 + 36 = 70.641016151
x2 = 70.641016151

Simplifying
x2 = 70.641016151

Take the square root of each side:
x = {-8.404821006, 8.404821006}


Subproblem 2
x2 + -36 = -34.641016151

Simplifying
x2 + -36 = -34.641016151

Reorder the terms:
-36 + x2 = -34.641016151

Solving
-36 + x2 = -34.641016151

Solving for variable 'x'.

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

Add '36' to each side of the equation.
-36 + 36 + x2 = -34.641016151 + 36

Combine like terms: -36 + 36 = 0
0 + x2 = -34.641016151 + 36
x2 = -34.641016151 + 36

Combine like terms: -34.641016151 + 36 = 1.358983849
x2 = 1.358983849

Simplifying
x2 = 1.358983849

Take the square root of each side:
x = {-1.165754626, 1.165754626}


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-8.404821006, 8.404821006, -1.165754626, 1.165754626}

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