2x^4-49x^2+54=0

Solution for 2x^4-49x^2+54=0 equation:

Simplifying
2x4 + -49x2 + 54 = 0

Reorder the terms:
54 + -49x2 + 2x4 = 0

Solving
54 + -49x2 + 2x4 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
27 + -24.5x2 + x4 = 0

Move the constant term to the right:

Add '-27' to each side of the equation.
27 + -24.5x2 + -27 + x4 = 0 + -27

Reorder the terms:
27 + -27 + -24.5x2 + x4 = 0 + -27

Combine like terms: 27 + -27 = 0
0 + -24.5x2 + x4 = 0 + -27
-24.5x2 + x4 = 0 + -27

Combine like terms: 0 + -27 = -27
-24.5x2 + x4 = -27

The x term is -24.5x2.  Take half its coefficient (-12.25).
Square it (150.0625) and add it to both sides.

Add '150.0625' to each side of the equation.
-24.5x2 + 150.0625 + x4 = -27 + 150.0625

Reorder the terms:
150.0625 + -24.5x2 + x4 = -27 + 150.0625

Combine like terms: -27 + 150.0625 = 123.0625
150.0625 + -24.5x2 + x4 = 123.0625

Factor a perfect square on the left side:
(x2 + -12.25)(x2 + -12.25) = 123.0625

Calculate the square root of the right side: 11.093353866

Break this problem into two subproblems by setting 
(x2 + -12.25) equal to 11.093353866 and -11.093353866.

Subproblem 1
x2 + -12.25 = 11.093353866

Simplifying
x2 + -12.25 = 11.093353866

Reorder the terms:
-12.25 + x2 = 11.093353866

Solving
-12.25 + x2 = 11.093353866

Solving for variable 'x'.

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

Add '12.25' to each side of the equation.
-12.25 + 12.25 + x2 = 11.093353866 + 12.25

Combine like terms: -12.25 + 12.25 = 0.00
0.00 + x2 = 11.093353866 + 12.25
x2 = 11.093353866 + 12.25

Combine like terms: 11.093353866 + 12.25 = 23.343353866
x2 = 23.343353866

Simplifying
x2 = 23.343353866

Take the square root of each side:
x = {-4.831496028, 4.831496028}

Subproblem 2
x2 + -12.25 = -11.093353866

Simplifying
x2 + -12.25 = -11.093353866

Reorder the terms:
-12.25 + x2 = -11.093353866

Solving
-12.25 + x2 = -11.093353866

Solving for variable 'x'.

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

Add '12.25' to each side of the equation.
-12.25 + 12.25 + x2 = -11.093353866 + 12.25

Combine like terms: -12.25 + 12.25 = 0.00
0.00 + x2 = -11.093353866 + 12.25
x2 = -11.093353866 + 12.25

Combine like terms: -11.093353866 + 12.25 = 1.156646134
x2 = 1.156646134

Simplifying
x2 = 1.156646134

Take the square root of each side:
x = {-1.075474841, 1.075474841}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-4.831496028, 4.831496028, -1.075474841, 1.075474841}