6x(x-1)=5-x

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Solution for 6x(x-1)=5-x equation:


Simplifying
6x(x + -1) = 5 + -1x

Reorder the terms:
6x(-1 + x) = 5 + -1x
(-1 * 6x + x * 6x) = 5 + -1x
(-6x + 6x2) = 5 + -1x

Solving
-6x + 6x2 = 5 + -1x

Solving for variable 'x'.

Reorder the terms:
-5 + -6x + x + 6x2 = 5 + -1x + -5 + x

Combine like terms: -6x + x = -5x
-5 + -5x + 6x2 = 5 + -1x + -5 + x

Reorder the terms:
-5 + -5x + 6x2 = 5 + -5 + -1x + x

Combine like terms: 5 + -5 = 0
-5 + -5x + 6x2 = 0 + -1x + x
-5 + -5x + 6x2 = -1x + x

Combine like terms: -1x + x = 0
-5 + -5x + 6x2 = 0

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

Divide each side by '6'.
-0.8333333333 + -0.8333333333x + x2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + -0.8333333333x + 0.8333333333 + x2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + -0.8333333333x + x2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + -0.8333333333x + x2 = 0 + 0.8333333333
-0.8333333333x + x2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
-0.8333333333x + x2 = 0.8333333333

The x term is -0.8333333333x.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333x + 0.1736111111 + x2 = 0.8333333333 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333x + x2 = 0.8333333333 + 0.1736111111

Combine like terms: 0.8333333333 + 0.1736111111 = 1.0069444444
0.1736111111 + -0.8333333333x + x2 = 1.0069444444

Factor a perfect square on the left side:
(x + -0.4166666667)(x + -0.4166666667) = 1.0069444444

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(x + -0.4166666667) equal to 1.003466215 and -1.003466215.

Subproblem 1

x + -0.4166666667 = 1.003466215 Simplifying x + -0.4166666667 = 1.003466215 Reorder the terms: -0.4166666667 + x = 1.003466215 Solving -0.4166666667 + x = 1.003466215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = 1.003466215 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 1.003466215 + 0.4166666667 x = 1.003466215 + 0.4166666667 Combine like terms: 1.003466215 + 0.4166666667 = 1.4201328817 x = 1.4201328817 Simplifying x = 1.4201328817

Subproblem 2

x + -0.4166666667 = -1.003466215 Simplifying x + -0.4166666667 = -1.003466215 Reorder the terms: -0.4166666667 + x = -1.003466215 Solving -0.4166666667 + x = -1.003466215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = -1.003466215 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -1.003466215 + 0.4166666667 x = -1.003466215 + 0.4166666667 Combine like terms: -1.003466215 + 0.4166666667 = -0.5867995483 x = -0.5867995483 Simplifying x = -0.5867995483

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.4201328817, -0.5867995483}

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