32-3x^2+9x+6=x^2-x-6

Solution for 32-3x^2+9x+6=x^2-x-6 equation:

Simplifying
32 + -3x2 + 9x + 6 = x2 + -1x + -6

Reorder the terms:
32 + 6 + 9x + -3x2 = x2 + -1x + -6

Combine like terms: 32 + 6 = 38
38 + 9x + -3x2 = x2 + -1x + -6

Reorder the terms:
38 + 9x + -3x2 = -6 + -1x + x2

Solving
38 + 9x + -3x2 = -6 + -1x + x2

Solving for variable 'x'.

Reorder the terms:
38 + 6 + 9x + x + -3x2 + -1x2 = -6 + -1x + x2 + 6 + x + -1x2

Combine like terms: 38 + 6 = 44
44 + 9x + x + -3x2 + -1x2 = -6 + -1x + x2 + 6 + x + -1x2

Combine like terms: 9x + x = 10x
44 + 10x + -3x2 + -1x2 = -6 + -1x + x2 + 6 + x + -1x2

Combine like terms: -3x2 + -1x2 = -4x2
44 + 10x + -4x2 = -6 + -1x + x2 + 6 + x + -1x2

Reorder the terms:
44 + 10x + -4x2 = -6 + 6 + -1x + x + x2 + -1x2

Combine like terms: -6 + 6 = 0
44 + 10x + -4x2 = 0 + -1x + x + x2 + -1x2
44 + 10x + -4x2 = -1x + x + x2 + -1x2

Combine like terms: -1x + x = 0
44 + 10x + -4x2 = 0 + x2 + -1x2
44 + 10x + -4x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
44 + 10x + -4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(22 + 5x + -2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(22 + 5x + -2x2)' equal to zero and attempt to solve:

Simplifying
22 + 5x + -2x2 = 0

Solving
22 + 5x + -2x2 = 0

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

Divide each side by '-2'.
-11 + -2.5x + x2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + -2.5x + 11 + x2 = 0 + 11

Reorder the terms:
-11 + 11 + -2.5x + x2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + -2.5x + x2 = 0 + 11
-2.5x + x2 = 0 + 11

Combine like terms: 0 + 11 = 11
-2.5x + x2 = 11

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 11 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 11 + 1.5625

Combine like terms: 11 + 1.5625 = 12.5625
1.5625 + -2.5x + x2 = 12.5625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 12.5625

Calculate the square root of the right side: 3.54436172

Break this problem into two subproblems by setting 
(x + -1.25) equal to 3.54436172 and -3.54436172.

Subproblem 1
x + -1.25 = 3.54436172

Simplifying
x + -1.25 = 3.54436172

Reorder the terms:
-1.25 + x = 3.54436172

Solving
-1.25 + x = 3.54436172

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = 3.54436172 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = 3.54436172 + 1.25
x = 3.54436172 + 1.25

Combine like terms: 3.54436172 + 1.25 = 4.79436172
x = 4.79436172

Simplifying
x = 4.79436172

Subproblem 2
x + -1.25 = -3.54436172

Simplifying
x + -1.25 = -3.54436172

Reorder the terms:
-1.25 + x = -3.54436172

Solving
-1.25 + x = -3.54436172

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = -3.54436172 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = -3.54436172 + 1.25
x = -3.54436172 + 1.25

Combine like terms: -3.54436172 + 1.25 = -2.29436172
x = -2.29436172

Simplifying
x = -2.29436172

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.79436172, -2.29436172}
Solution
x = {4.79436172, -2.29436172}