6x(x-1)=6x+1

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Solution for 6x(x-1)=6x+1 equation:


Simplifying
6x(x + -1) = 6x + 1

Reorder the terms:
6x(-1 + x) = 6x + 1
(-1 * 6x + x * 6x) = 6x + 1
(-6x + 6x2) = 6x + 1

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

Solving
-6x + 6x2 = 1 + 6x

Solving for variable 'x'.

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

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

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

Combine like terms: 1 + -1 = 0
-1 + -12x + 6x2 = 0 + 6x + -6x
-1 + -12x + 6x2 = 6x + -6x

Combine like terms: 6x + -6x = 0
-1 + -12x + 6x2 = 0

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

Divide each side by '6'.
-0.1666666667 + -2x + x2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -2x + 0.1666666667 + x2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -2x + x2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -2x + x2 = 0 + 0.1666666667
-2x + x2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-2x + x2 = 0.1666666667

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 0.1666666667 + 1

Reorder the terms:
1 + -2x + x2 = 0.1666666667 + 1

Combine like terms: 0.1666666667 + 1 = 1.1666666667
1 + -2x + x2 = 1.1666666667

Factor a perfect square on the left side:
(x + -1)(x + -1) = 1.1666666667

Calculate the square root of the right side: 1.08012345

Break this problem into two subproblems by setting 
(x + -1) equal to 1.08012345 and -1.08012345.

Subproblem 1

x + -1 = 1.08012345 Simplifying x + -1 = 1.08012345 Reorder the terms: -1 + x = 1.08012345 Solving -1 + x = 1.08012345 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.08012345 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.08012345 + 1 x = 1.08012345 + 1 Combine like terms: 1.08012345 + 1 = 2.08012345 x = 2.08012345 Simplifying x = 2.08012345

Subproblem 2

x + -1 = -1.08012345 Simplifying x + -1 = -1.08012345 Reorder the terms: -1 + x = -1.08012345 Solving -1 + x = -1.08012345 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.08012345 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.08012345 + 1 x = -1.08012345 + 1 Combine like terms: -1.08012345 + 1 = -0.08012345 x = -0.08012345 Simplifying x = -0.08012345

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.08012345, -0.08012345}

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