188-2x^2=x^2+33x+17

Solution for 188-2x^2=x^2+33x+17 equation:

Simplifying
188 + -2x2 = x2 + 33x + 17

Reorder the terms:
188 + -2x2 = 17 + 33x + x2

Solving
188 + -2x2 = 17 + 33x + x2

Solving for variable 'x'.

Reorder the terms:
188 + -17 + -33x + -2x2 + -1x2 = 17 + 33x + x2 + -17 + -33x + -1x2

Combine like terms: 188 + -17 = 171
171 + -33x + -2x2 + -1x2 = 17 + 33x + x2 + -17 + -33x + -1x2

Combine like terms: -2x2 + -1x2 = -3x2
171 + -33x + -3x2 = 17 + 33x + x2 + -17 + -33x + -1x2

Reorder the terms:
171 + -33x + -3x2 = 17 + -17 + 33x + -33x + x2 + -1x2

Combine like terms: 17 + -17 = 0
171 + -33x + -3x2 = 0 + 33x + -33x + x2 + -1x2
171 + -33x + -3x2 = 33x + -33x + x2 + -1x2

Combine like terms: 33x + -33x = 0
171 + -33x + -3x2 = 0 + x2 + -1x2
171 + -33x + -3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
171 + -33x + -3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(57 + -11x + -1x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(57 + -11x + -1x2)' equal to zero and attempt to solve:

Simplifying
57 + -11x + -1x2 = 0

Solving
57 + -11x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-57 + 11x + x2 = 0

Move the constant term to the right:

Add '57' to each side of the equation.
-57 + 11x + 57 + x2 = 0 + 57

Reorder the terms:
-57 + 57 + 11x + x2 = 0 + 57

Combine like terms: -57 + 57 = 0
0 + 11x + x2 = 0 + 57
11x + x2 = 0 + 57

Combine like terms: 0 + 57 = 57
11x + x2 = 57

The x term is 11x.  Take half its coefficient (5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
11x + 30.25 + x2 = 57 + 30.25

Reorder the terms:
30.25 + 11x + x2 = 57 + 30.25

Combine like terms: 57 + 30.25 = 87.25
30.25 + 11x + x2 = 87.25

Factor a perfect square on the left side:
(x + 5.5)(x + 5.5) = 87.25

Calculate the square root of the right side: 9.340770846

Break this problem into two subproblems by setting 
(x + 5.5) equal to 9.340770846 and -9.340770846.

Subproblem 1
x + 5.5 = 9.340770846

Simplifying
x + 5.5 = 9.340770846

Reorder the terms:
5.5 + x = 9.340770846

Solving
5.5 + x = 9.340770846

Solving for variable 'x'.

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

Add '-5.5' to each side of the equation.
5.5 + -5.5 + x = 9.340770846 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + x = 9.340770846 + -5.5
x = 9.340770846 + -5.5

Combine like terms: 9.340770846 + -5.5 = 3.840770846
x = 3.840770846

Simplifying
x = 3.840770846

Subproblem 2
x + 5.5 = -9.340770846

Simplifying
x + 5.5 = -9.340770846

Reorder the terms:
5.5 + x = -9.340770846

Solving
5.5 + x = -9.340770846

Solving for variable 'x'.

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

Add '-5.5' to each side of the equation.
5.5 + -5.5 + x = -9.340770846 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + x = -9.340770846 + -5.5
x = -9.340770846 + -5.5

Combine like terms: -9.340770846 + -5.5 = -14.840770846
x = -14.840770846

Simplifying
x = -14.840770846

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.840770846, -14.840770846}
Solution
x = {3.840770846, -14.840770846}