6x(3x+1)=1

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Solution for 6x(3x+1)=1 equation:


Simplifying
6x(3x + 1) = 1

Reorder the terms:
6x(1 + 3x) = 1
(1 * 6x + 3x * 6x) = 1
(6x + 18x2) = 1

Solving
6x + 18x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 6x + 18x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 6x + 18x2 = 0

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

Divide each side by '18'.
-0.05555555556 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.05555555556' to each side of the equation.
-0.05555555556 + 0.3333333333x + 0.05555555556 + x2 = 0 + 0.05555555556

Reorder the terms:
-0.05555555556 + 0.05555555556 + 0.3333333333x + x2 = 0 + 0.05555555556

Combine like terms: -0.05555555556 + 0.05555555556 = 0.00000000000
0.00000000000 + 0.3333333333x + x2 = 0 + 0.05555555556
0.3333333333x + x2 = 0 + 0.05555555556

Combine like terms: 0 + 0.05555555556 = 0.05555555556
0.3333333333x + x2 = 0.05555555556

The x term is 0.3333333333x.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333x + 0.02777777779 + x2 = 0.05555555556 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 0.05555555556 + 0.02777777779

Combine like terms: 0.05555555556 + 0.02777777779 = 0.08333333335
0.02777777779 + 0.3333333333x + x2 = 0.08333333335

Factor a perfect square on the left side:
(x + 0.1666666667)(x + 0.1666666667) = 0.08333333335

Calculate the square root of the right side: 0.288675135

Break this problem into two subproblems by setting 
(x + 0.1666666667) equal to 0.288675135 and -0.288675135.

Subproblem 1

x + 0.1666666667 = 0.288675135 Simplifying x + 0.1666666667 = 0.288675135 Reorder the terms: 0.1666666667 + x = 0.288675135 Solving 0.1666666667 + x = 0.288675135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 0.288675135 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 0.288675135 + -0.1666666667 x = 0.288675135 + -0.1666666667 Combine like terms: 0.288675135 + -0.1666666667 = 0.1220084683 x = 0.1220084683 Simplifying x = 0.1220084683

Subproblem 2

x + 0.1666666667 = -0.288675135 Simplifying x + 0.1666666667 = -0.288675135 Reorder the terms: 0.1666666667 + x = -0.288675135 Solving 0.1666666667 + x = -0.288675135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -0.288675135 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -0.288675135 + -0.1666666667 x = -0.288675135 + -0.1666666667 Combine like terms: -0.288675135 + -0.1666666667 = -0.4553418017 x = -0.4553418017 Simplifying x = -0.4553418017

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.1220084683, -0.4553418017}

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