6x-5=5x(x-3)+x

Solution for 6x-5=5x(x-3)+x equation:

Simplifying
6x + -5 = 5x(x + -3) + x

Reorder the terms:
-5 + 6x = 5x(x + -3) + x

Reorder the terms:
-5 + 6x = 5x(-3 + x) + x
-5 + 6x = (-3 * 5x + x * 5x) + x
-5 + 6x = (-15x + 5x2) + x

Reorder the terms:
-5 + 6x = -15x + x + 5x2

Combine like terms: -15x + x = -14x
-5 + 6x = -14x + 5x2

Solving
-5 + 6x = -14x + 5x2

Solving for variable 'x'.

Combine like terms: 6x + 14x = 20x
-5 + 20x + -5x2 = -14x + 5x2 + 14x + -5x2

Reorder the terms:
-5 + 20x + -5x2 = -14x + 14x + 5x2 + -5x2

Combine like terms: -14x + 14x = 0
-5 + 20x + -5x2 = 0 + 5x2 + -5x2
-5 + 20x + -5x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
-5 + 20x + -5x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-1 + 4x + -1x2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-1 + 4x + -1x2)' equal to zero and attempt to solve:

Simplifying
-1 + 4x + -1x2 = 0

Solving
-1 + 4x + -1x2 = 0

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

Divide each side by '-1'.
1 + -4x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -4x + x2 = 0 + -1

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

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

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = -1 + 4

Reorder the terms:
4 + -4x + x2 = -1 + 4

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

Factor a perfect square on the left side:
(x + -2)(x + -2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + -2) equal to 1.732050808 and -1.732050808.

Subproblem 1
x + -2 = 1.732050808

Simplifying
x + -2 = 1.732050808

Reorder the terms:
-2 + x = 1.732050808

Solving
-2 + x = 1.732050808

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = 1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + x = 1.732050808 + 2
x = 1.732050808 + 2

Combine like terms: 1.732050808 + 2 = 3.732050808
x = 3.732050808

Simplifying
x = 3.732050808

Subproblem 2
x + -2 = -1.732050808

Simplifying
x + -2 = -1.732050808

Reorder the terms:
-2 + x = -1.732050808

Solving
-2 + x = -1.732050808

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = -1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + x = -1.732050808 + 2
x = -1.732050808 + 2

Combine like terms: -1.732050808 + 2 = 0.267949192
x = 0.267949192

Simplifying
x = 0.267949192

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.732050808, 0.267949192}
Solution
x = {3.732050808, 0.267949192}