4(x-3)x=x-6

Solution for 4(x-3)x=x-6 equation:

Simplifying
4(x + -3) * x = x + -6

Reorder the terms:
4(-3 + x) * x = x + -6

Reorder the terms for easier multiplication:
4x(-3 + x) = x + -6
(-3 * 4x + x * 4x) = x + -6
(-12x + 4x2) = x + -6

Reorder the terms:
-12x + 4x2 = -6 + x

Solving
-12x + 4x2 = -6 + x

Solving for variable 'x'.

Reorder the terms:
6 + -12x + -1x + 4x2 = -6 + x + 6 + -1x

Combine like terms: -12x + -1x = -13x
6 + -13x + 4x2 = -6 + x + 6 + -1x

Reorder the terms:
6 + -13x + 4x2 = -6 + 6 + x + -1x

Combine like terms: -6 + 6 = 0
6 + -13x + 4x2 = 0 + x + -1x
6 + -13x + 4x2 = x + -1x

Combine like terms: x + -1x = 0
6 + -13x + 4x2 = 0

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

Divide each side by '4'.
1.5 + -3.25x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3.25x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3.25x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3.25x + x2 = 0 + -1.5
-3.25x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3.25x + x2 = -1.5

The x term is -3.25x.  Take half its coefficient (-1.625).
Square it (2.640625) and add it to both sides.

Add '2.640625' to each side of the equation.
-3.25x + 2.640625 + x2 = -1.5 + 2.640625

Reorder the terms:
2.640625 + -3.25x + x2 = -1.5 + 2.640625

Combine like terms: -1.5 + 2.640625 = 1.140625
2.640625 + -3.25x + x2 = 1.140625

Factor a perfect square on the left side:
(x + -1.625)(x + -1.625) = 1.140625

Calculate the square root of the right side: 1.068000468

Break this problem into two subproblems by setting 
(x + -1.625) equal to 1.068000468 and -1.068000468.

Subproblem 1
x + -1.625 = 1.068000468

Simplifying
x + -1.625 = 1.068000468

Reorder the terms:
-1.625 + x = 1.068000468

Solving
-1.625 + x = 1.068000468

Solving for variable 'x'.

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

Add '1.625' to each side of the equation.
-1.625 + 1.625 + x = 1.068000468 + 1.625

Combine like terms: -1.625 + 1.625 = 0.000
0.000 + x = 1.068000468 + 1.625
x = 1.068000468 + 1.625

Combine like terms: 1.068000468 + 1.625 = 2.693000468
x = 2.693000468

Simplifying
x = 2.693000468

Subproblem 2
x + -1.625 = -1.068000468

Simplifying
x + -1.625 = -1.068000468

Reorder the terms:
-1.625 + x = -1.068000468

Solving
-1.625 + x = -1.068000468

Solving for variable 'x'.

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

Add '1.625' to each side of the equation.
-1.625 + 1.625 + x = -1.068000468 + 1.625

Combine like terms: -1.625 + 1.625 = 0.000
0.000 + x = -1.068000468 + 1.625
x = -1.068000468 + 1.625

Combine like terms: -1.068000468 + 1.625 = 0.556999532
x = 0.556999532

Simplifying
x = 0.556999532

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.693000468, 0.556999532}