3x(2x+1)=1

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Solution for 3x(2x+1)=1 equation:


Simplifying
3x(2x + 1) = 1

Reorder the terms:
3x(1 + 2x) = 1
(1 * 3x + 2x * 3x) = 1
(3x + 6x2) = 1

Solving
3x + 6x2 = 1

Solving for variable 'x'.

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

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

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

Divide each side by '6'.
-0.1666666667 + 0.5x + x2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.5x + 0.1666666667 + x2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 0.5x + x2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 0.5x + x2 = 0 + 0.1666666667
0.5x + x2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
0.5x + x2 = 0.1666666667

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 0.1666666667 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 0.1666666667 + 0.0625

Combine like terms: 0.1666666667 + 0.0625 = 0.2291666667
0.0625 + 0.5x + x2 = 0.2291666667

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 0.2291666667

Calculate the square root of the right side: 0.478713554

Break this problem into two subproblems by setting 
(x + 0.25) equal to 0.478713554 and -0.478713554.

Subproblem 1

x + 0.25 = 0.478713554 Simplifying x + 0.25 = 0.478713554 Reorder the terms: 0.25 + x = 0.478713554 Solving 0.25 + x = 0.478713554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.478713554 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.478713554 + -0.25 x = 0.478713554 + -0.25 Combine like terms: 0.478713554 + -0.25 = 0.228713554 x = 0.228713554 Simplifying x = 0.228713554

Subproblem 2

x + 0.25 = -0.478713554 Simplifying x + 0.25 = -0.478713554 Reorder the terms: 0.25 + x = -0.478713554 Solving 0.25 + x = -0.478713554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.478713554 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.478713554 + -0.25 x = -0.478713554 + -0.25 Combine like terms: -0.478713554 + -0.25 = -0.728713554 x = -0.728713554 Simplifying x = -0.728713554

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.228713554, -0.728713554}

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