x(3x+1)=6

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


Simplifying
x(3x + 1) = 6

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

Solving
1x + 3x2 = 6

Solving for variable 'x'.

Reorder the terms:
-6 + 1x + 3x2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 1x + 3x2 = 0

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

Divide each side by '3'.
-2 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.3333333333x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.3333333333x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.3333333333x + x2 = 0 + 2
0.3333333333x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.3333333333x + x2 = 2

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 = 2 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 2 + 0.02777777779

Combine like terms: 2 + 0.02777777779 = 2.02777777779
0.02777777779 + 0.3333333333x + x2 = 2.02777777779

Factor a perfect square on the left side:
(x + 0.1666666667)(x + 0.1666666667) = 2.02777777779

Calculate the square root of the right side: 1.424000624

Break this problem into two subproblems by setting 
(x + 0.1666666667) equal to 1.424000624 and -1.424000624.

Subproblem 1

x + 0.1666666667 = 1.424000624 Simplifying x + 0.1666666667 = 1.424000624 Reorder the terms: 0.1666666667 + x = 1.424000624 Solving 0.1666666667 + x = 1.424000624 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 = 1.424000624 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 1.424000624 + -0.1666666667 x = 1.424000624 + -0.1666666667 Combine like terms: 1.424000624 + -0.1666666667 = 1.2573339573 x = 1.2573339573 Simplifying x = 1.2573339573

Subproblem 2

x + 0.1666666667 = -1.424000624 Simplifying x + 0.1666666667 = -1.424000624 Reorder the terms: 0.1666666667 + x = -1.424000624 Solving 0.1666666667 + x = -1.424000624 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 = -1.424000624 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -1.424000624 + -0.1666666667 x = -1.424000624 + -0.1666666667 Combine like terms: -1.424000624 + -0.1666666667 = -1.5906672907 x = -1.5906672907 Simplifying x = -1.5906672907

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.2573339573, -1.5906672907}

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