3x(3x-2)=1

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


Simplifying
3x(3x + -2) = 1

Reorder the terms:
3x(-2 + 3x) = 1
(-2 * 3x + 3x * 3x) = 1
(-6x + 9x2) = 1

Solving
-6x + 9x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -6x + 9x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -6x + 9x2 = 0

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

Divide each side by '9'.
-0.1111111111 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.6666666667x + 0.1111111111 + x2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.6666666667x + x2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.6666666667x + x2 = 0 + 0.1111111111
-0.6666666667x + x2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.6666666667x + x2 = 0.1111111111

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 0.1111111111 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 0.1111111111 + 0.1111111112

Combine like terms: 0.1111111111 + 0.1111111112 = 0.2222222223
0.1111111112 + -0.6666666667x + x2 = 0.2222222223

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 0.2222222223

Calculate the square root of the right side: 0.471404521

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 0.471404521 and -0.471404521.

Subproblem 1

x + -0.3333333334 = 0.471404521 Simplifying x + -0.3333333334 = 0.471404521 Reorder the terms: -0.3333333334 + x = 0.471404521 Solving -0.3333333334 + x = 0.471404521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 0.471404521 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 0.471404521 + 0.3333333334 x = 0.471404521 + 0.3333333334 Combine like terms: 0.471404521 + 0.3333333334 = 0.8047378544 x = 0.8047378544 Simplifying x = 0.8047378544

Subproblem 2

x + -0.3333333334 = -0.471404521 Simplifying x + -0.3333333334 = -0.471404521 Reorder the terms: -0.3333333334 + x = -0.471404521 Solving -0.3333333334 + x = -0.471404521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -0.471404521 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -0.471404521 + 0.3333333334 x = -0.471404521 + 0.3333333334 Combine like terms: -0.471404521 + 0.3333333334 = -0.1380711876 x = -0.1380711876 Simplifying x = -0.1380711876

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.8047378544, -0.1380711876}

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