9x=(3x-1)(x-1)

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Solution for 9x=(3x-1)(x-1) equation: 


Simplifying
9x = (3x + -1)(x + -1)

Reorder the terms:
9x = (-1 + 3x)(x + -1)

Reorder the terms:
9x = (-1 + 3x)(-1 + x)

Multiply (-1 + 3x) * (-1 + x)
9x = (-1(-1 + x) + 3x * (-1 + x))
9x = ((-1 * -1 + x * -1) + 3x * (-1 + x))
9x = ((1 + -1x) + 3x * (-1 + x))
9x = (1 + -1x + (-1 * 3x + x * 3x))
9x = (1 + -1x + (-3x + 3x2))

Combine like terms: -1x + -3x = -4x
9x = (1 + -4x + 3x2)

Solving
9x = 1 + -4x + 3x2

Solving for variable 'x'.

Reorder the terms:
-1 + 9x + 4x + -3x2 = 1 + -4x + 3x2 + -1 + 4x + -3x2

Combine like terms: 9x + 4x = 13x
-1 + 13x + -3x2 = 1 + -4x + 3x2 + -1 + 4x + -3x2

Reorder the terms:
-1 + 13x + -3x2 = 1 + -1 + -4x + 4x + 3x2 + -3x2

Combine like terms: 1 + -1 = 0
-1 + 13x + -3x2 = 0 + -4x + 4x + 3x2 + -3x2
-1 + 13x + -3x2 = -4x + 4x + 3x2 + -3x2

Combine like terms: -4x + 4x = 0
-1 + 13x + -3x2 = 0 + 3x2 + -3x2
-1 + 13x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-1 + 13x + -3x2 = 0

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

Divide each side by '-3'.
0.3333333333 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -4.333333333x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -4.333333333x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -4.333333333x + x2 = 0 + -0.3333333333
-4.333333333x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-4.333333333x + x2 = -0.3333333333

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = -0.3333333333 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = -0.3333333333 + 4.694444446

Combine like terms: -0.3333333333 + 4.694444446 = 4.3611111127
4.694444446 + -4.333333333x + x2 = 4.3611111127

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 4.3611111127

Calculate the square root of the right side: 2.088327348

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 2.088327348 and -2.088327348.


Subproblem 1
x + -2.166666667 = 2.088327348

Simplifying
x + -2.166666667 = 2.088327348

Reorder the terms:
-2.166666667 + x = 2.088327348

Solving
-2.166666667 + x = 2.088327348

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = 2.088327348 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 2.088327348 + 2.166666667
x = 2.088327348 + 2.166666667

Combine like terms: 2.088327348 + 2.166666667 = 4.254994015
x = 4.254994015

Simplifying
x = 4.254994015


Subproblem 2
x + -2.166666667 = -2.088327348

Simplifying
x + -2.166666667 = -2.088327348

Reorder the terms:
-2.166666667 + x = -2.088327348

Solving
-2.166666667 + x = -2.088327348

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = -2.088327348 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -2.088327348 + 2.166666667
x = -2.088327348 + 2.166666667

Combine like terms: -2.088327348 + 2.166666667 = 0.078339319
x = 0.078339319

Simplifying
x = 0.078339319


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.254994015, 0.078339319}

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