x-7=x(56+77x)

Solution for x-7=x(56+77x) equation:

Simplifying
x + -7 = x(56 + 77x)

Reorder the terms:
-7 + x = x(56 + 77x)
-7 + x = (56 * x + 77x * x)
-7 + x = (56x + 77x2)

Solving
-7 + x = 56x + 77x2

Solving for variable 'x'.

Combine like terms: x + -56x = -55x
-7 + -55x + -77x2 = 56x + 77x2 + -56x + -77x2

Reorder the terms:
-7 + -55x + -77x2 = 56x + -56x + 77x2 + -77x2

Combine like terms: 56x + -56x = 0
-7 + -55x + -77x2 = 0 + 77x2 + -77x2
-7 + -55x + -77x2 = 77x2 + -77x2

Combine like terms: 77x2 + -77x2 = 0
-7 + -55x + -77x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(7 + 55x + 77x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(7 + 55x + 77x2)' equal to zero and attempt to solve:

Simplifying
7 + 55x + 77x2 = 0

Solving
7 + 55x + 77x2 = 0

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

Divide each side by '77'.
0.09090909091 + 0.7142857143x + x2 = 0

Move the constant term to the right:

Add '-0.09090909091' to each side of the equation.
0.09090909091 + 0.7142857143x + -0.09090909091 + x2 = 0 + -0.09090909091

Reorder the terms:
0.09090909091 + -0.09090909091 + 0.7142857143x + x2 = 0 + -0.09090909091

Combine like terms: 0.09090909091 + -0.09090909091 = 0.00000000000
0.00000000000 + 0.7142857143x + x2 = 0 + -0.09090909091
0.7142857143x + x2 = 0 + -0.09090909091

Combine like terms: 0 + -0.09090909091 = -0.09090909091
0.7142857143x + x2 = -0.09090909091

The x term is 0.7142857143x.  Take half its coefficient (0.3571428572).
Square it (0.1275510204) and add it to both sides.

Add '0.1275510204' to each side of the equation.
0.7142857143x + 0.1275510204 + x2 = -0.09090909091 + 0.1275510204

Reorder the terms:
0.1275510204 + 0.7142857143x + x2 = -0.09090909091 + 0.1275510204

Combine like terms: -0.09090909091 + 0.1275510204 = 0.03664192949
0.1275510204 + 0.7142857143x + x2 = 0.03664192949

Factor a perfect square on the left side:
(x + 0.3571428572)(x + 0.3571428572) = 0.03664192949

Calculate the square root of the right side: 0.191420818

Break this problem into two subproblems by setting 
(x + 0.3571428572) equal to 0.191420818 and -0.191420818.

Subproblem 1
x + 0.3571428572 = 0.191420818

Simplifying
x + 0.3571428572 = 0.191420818

Reorder the terms:
0.3571428572 + x = 0.191420818

Solving
0.3571428572 + x = 0.191420818

Solving for variable 'x'.

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

Add '-0.3571428572' to each side of the equation.
0.3571428572 + -0.3571428572 + x = 0.191420818 + -0.3571428572

Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000
0.0000000000 + x = 0.191420818 + -0.3571428572
x = 0.191420818 + -0.3571428572

Combine like terms: 0.191420818 + -0.3571428572 = -0.1657220392
x = -0.1657220392

Simplifying
x = -0.1657220392

Subproblem 2
x + 0.3571428572 = -0.191420818

Simplifying
x + 0.3571428572 = -0.191420818

Reorder the terms:
0.3571428572 + x = -0.191420818

Solving
0.3571428572 + x = -0.191420818

Solving for variable 'x'.

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

Add '-0.3571428572' to each side of the equation.
0.3571428572 + -0.3571428572 + x = -0.191420818 + -0.3571428572

Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000
0.0000000000 + x = -0.191420818 + -0.3571428572
x = -0.191420818 + -0.3571428572

Combine like terms: -0.191420818 + -0.3571428572 = -0.5485636752
x = -0.5485636752

Simplifying
x = -0.5485636752

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.1657220392, -0.5485636752}
Solution
x = {-0.1657220392, -0.5485636752}