7x+6=1x+8x+6x^2+6x

Solution for 7x+6=1x+8x+6x^2+6x equation:

Simplifying
7x + 6 = 1x + 8x + 6x2 + 6x

Reorder the terms:
6 + 7x = 1x + 8x + 6x2 + 6x

Reorder the terms:
6 + 7x = 1x + 8x + 6x + 6x2

Combine like terms: 1x + 8x = 9x
6 + 7x = 9x + 6x + 6x2

Combine like terms: 9x + 6x = 15x
6 + 7x = 15x + 6x2

Solving
6 + 7x = 15x + 6x2

Solving for variable 'x'.

Combine like terms: 7x + -15x = -8x
6 + -8x + -6x2 = 15x + 6x2 + -15x + -6x2

Reorder the terms:
6 + -8x + -6x2 = 15x + -15x + 6x2 + -6x2

Combine like terms: 15x + -15x = 0
6 + -8x + -6x2 = 0 + 6x2 + -6x2
6 + -8x + -6x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
6 + -8x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + -4x + -3x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + -4x + -3x2)' equal to zero and attempt to solve:

Simplifying
3 + -4x + -3x2 = 0

Solving
3 + -4x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-1 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.333333333x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.333333333x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.333333333x + x2 = 0 + 1
1.333333333x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.333333333x + x2 = 1

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 1 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 1 + 0.4444444442

Combine like terms: 1 + 0.4444444442 = 1.4444444442
0.4444444442 + 1.333333333x + x2 = 1.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 1.4444444442

Calculate the square root of the right side: 1.201850425

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 1.201850425 and -1.201850425.

Subproblem 1
x + 0.6666666665 = 1.201850425

Simplifying
x + 0.6666666665 = 1.201850425

Reorder the terms:
0.6666666665 + x = 1.201850425

Solving
0.6666666665 + x = 1.201850425

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = 1.201850425 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = 1.201850425 + -0.6666666665
x = 1.201850425 + -0.6666666665

Combine like terms: 1.201850425 + -0.6666666665 = 0.5351837585
x = 0.5351837585

Simplifying
x = 0.5351837585

Subproblem 2
x + 0.6666666665 = -1.201850425

Simplifying
x + 0.6666666665 = -1.201850425

Reorder the terms:
0.6666666665 + x = -1.201850425

Solving
0.6666666665 + x = -1.201850425

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = -1.201850425 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = -1.201850425 + -0.6666666665
x = -1.201850425 + -0.6666666665

Combine like terms: -1.201850425 + -0.6666666665 = -1.8685170915
x = -1.8685170915

Simplifying
x = -1.8685170915

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5351837585, -1.8685170915}
Solution
x = {0.5351837585, -1.8685170915}