4x^4-97x^2+36=0

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Solution for 4x^4-97x^2+36=0 equation: 


Simplifying
4x4 + -97x2 + 36 = 0

Reorder the terms:
36 + -97x2 + 4x4 = 0

Solving
36 + -97x2 + 4x4 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
9 + -24.25x2 + x4 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -24.25x2 + -9 + x4 = 0 + -9

Reorder the terms:
9 + -9 + -24.25x2 + x4 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -24.25x2 + x4 = 0 + -9
-24.25x2 + x4 = 0 + -9

Combine like terms: 0 + -9 = -9
-24.25x2 + x4 = -9

The x term is -24.25x2.  Take half its coefficient (-12.125).
Square it (147.015625) and add it to both sides.

Add '147.015625' to each side of the equation.
-24.25x2 + 147.015625 + x4 = -9 + 147.015625

Reorder the terms:
147.015625 + -24.25x2 + x4 = -9 + 147.015625

Combine like terms: -9 + 147.015625 = 138.015625
147.015625 + -24.25x2 + x4 = 138.015625

Factor a perfect square on the left side:
(x2 + -12.125)(x2 + -12.125) = 138.015625

Calculate the square root of the right side: 11.74800515

Break this problem into two subproblems by setting 
(x2 + -12.125) equal to 11.74800515 and -11.74800515.


Subproblem 1
x2 + -12.125 = 11.74800515

Simplifying
x2 + -12.125 = 11.74800515

Reorder the terms:
-12.125 + x2 = 11.74800515

Solving
-12.125 + x2 = 11.74800515

Solving for variable 'x'.

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

Add '12.125' to each side of the equation.
-12.125 + 12.125 + x2 = 11.74800515 + 12.125

Combine like terms: -12.125 + 12.125 = 0.000
0.000 + x2 = 11.74800515 + 12.125
x2 = 11.74800515 + 12.125

Combine like terms: 11.74800515 + 12.125 = 23.87300515
x2 = 23.87300515

Simplifying
x2 = 23.87300515

Take the square root of each side:
x = {-4.886000936, 4.886000936}


Subproblem 2
x2 + -12.125 = -11.74800515

Simplifying
x2 + -12.125 = -11.74800515

Reorder the terms:
-12.125 + x2 = -11.74800515

Solving
-12.125 + x2 = -11.74800515

Solving for variable 'x'.

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

Add '12.125' to each side of the equation.
-12.125 + 12.125 + x2 = -11.74800515 + 12.125

Combine like terms: -12.125 + 12.125 = 0.000
0.000 + x2 = -11.74800515 + 12.125
x2 = -11.74800515 + 12.125

Combine like terms: -11.74800515 + 12.125 = 0.37699485
x2 = 0.37699485

Simplifying
x2 = 0.37699485

Take the square root of each side:
x = {-0.613999064, 0.613999064}


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-4.886000936, 4.886000936, -0.613999064, 0.613999064}

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