216x^2-144x-22=6x^2+54

Solution for 216x^2-144x-22=6x^2+54 equation:

Simplifying
216x2 + -144x + -22 = 6x2 + 54

Reorder the terms:
-22 + -144x + 216x2 = 6x2 + 54

Reorder the terms:
-22 + -144x + 216x2 = 54 + 6x2

Solving
-22 + -144x + 216x2 = 54 + 6x2

Solving for variable 'x'.

Reorder the terms:
-22 + -54 + -144x + 216x2 + -6x2 = 54 + 6x2 + -54 + -6x2

Combine like terms: -22 + -54 = -76
-76 + -144x + 216x2 + -6x2 = 54 + 6x2 + -54 + -6x2

Combine like terms: 216x2 + -6x2 = 210x2
-76 + -144x + 210x2 = 54 + 6x2 + -54 + -6x2

Reorder the terms:
-76 + -144x + 210x2 = 54 + -54 + 6x2 + -6x2

Combine like terms: 54 + -54 = 0
-76 + -144x + 210x2 = 0 + 6x2 + -6x2
-76 + -144x + 210x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
-76 + -144x + 210x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-38 + -72x + 105x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-38 + -72x + 105x2)' equal to zero and attempt to solve:

Simplifying
-38 + -72x + 105x2 = 0

Solving
-38 + -72x + 105x2 = 0

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

Divide each side by '105'.
-0.3619047619 + -0.6857142857x + x2 = 0

Move the constant term to the right:

Add '0.3619047619' to each side of the equation.
-0.3619047619 + -0.6857142857x + 0.3619047619 + x2 = 0 + 0.3619047619

Reorder the terms:
-0.3619047619 + 0.3619047619 + -0.6857142857x + x2 = 0 + 0.3619047619

Combine like terms: -0.3619047619 + 0.3619047619 = 0.0000000000
0.0000000000 + -0.6857142857x + x2 = 0 + 0.3619047619
-0.6857142857x + x2 = 0 + 0.3619047619

Combine like terms: 0 + 0.3619047619 = 0.3619047619
-0.6857142857x + x2 = 0.3619047619

The x term is -0.6857142857x.  Take half its coefficient (-0.3428571429).
Square it (0.1175510204) and add it to both sides.

Add '0.1175510204' to each side of the equation.
-0.6857142857x + 0.1175510204 + x2 = 0.3619047619 + 0.1175510204

Reorder the terms:
0.1175510204 + -0.6857142857x + x2 = 0.3619047619 + 0.1175510204

Combine like terms: 0.3619047619 + 0.1175510204 = 0.4794557823
0.1175510204 + -0.6857142857x + x2 = 0.4794557823

Factor a perfect square on the left side:
(x + -0.3428571429)(x + -0.3428571429) = 0.4794557823

Calculate the square root of the right side: 0.692427456

Break this problem into two subproblems by setting 
(x + -0.3428571429) equal to 0.692427456 and -0.692427456.

Subproblem 1
x + -0.3428571429 = 0.692427456

Simplifying
x + -0.3428571429 = 0.692427456

Reorder the terms:
-0.3428571429 + x = 0.692427456

Solving
-0.3428571429 + x = 0.692427456

Solving for variable 'x'.

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

Add '0.3428571429' to each side of the equation.
-0.3428571429 + 0.3428571429 + x = 0.692427456 + 0.3428571429

Combine like terms: -0.3428571429 + 0.3428571429 = 0.0000000000
0.0000000000 + x = 0.692427456 + 0.3428571429
x = 0.692427456 + 0.3428571429

Combine like terms: 0.692427456 + 0.3428571429 = 1.0352845989
x = 1.0352845989

Simplifying
x = 1.0352845989

Subproblem 2
x + -0.3428571429 = -0.692427456

Simplifying
x + -0.3428571429 = -0.692427456

Reorder the terms:
-0.3428571429 + x = -0.692427456

Solving
-0.3428571429 + x = -0.692427456

Solving for variable 'x'.

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

Add '0.3428571429' to each side of the equation.
-0.3428571429 + 0.3428571429 + x = -0.692427456 + 0.3428571429

Combine like terms: -0.3428571429 + 0.3428571429 = 0.0000000000
0.0000000000 + x = -0.692427456 + 0.3428571429
x = -0.692427456 + 0.3428571429

Combine like terms: -0.692427456 + 0.3428571429 = -0.3495703131
x = -0.3495703131

Simplifying
x = -0.3495703131

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.0352845989, -0.3495703131}
Solution
x = {1.0352845989, -0.3495703131}