(3x-1)(3x-1)=18

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Solution for (3x-1)(3x-1)=18 equation:


Simplifying
(3x + -1)(3x + -1) = 18

Reorder the terms:
(-1 + 3x)(3x + -1) = 18

Reorder the terms:
(-1 + 3x)(-1 + 3x) = 18

Multiply (-1 + 3x) * (-1 + 3x)
(-1(-1 + 3x) + 3x * (-1 + 3x)) = 18
((-1 * -1 + 3x * -1) + 3x * (-1 + 3x)) = 18
((1 + -3x) + 3x * (-1 + 3x)) = 18
(1 + -3x + (-1 * 3x + 3x * 3x)) = 18
(1 + -3x + (-3x + 9x2)) = 18

Combine like terms: -3x + -3x = -6x
(1 + -6x + 9x2) = 18

Solving
1 + -6x + 9x2 = 18

Solving for variable 'x'.

Reorder the terms:
1 + -18 + -6x + 9x2 = 18 + -18

Combine like terms: 1 + -18 = -17
-17 + -6x + 9x2 = 18 + -18

Combine like terms: 18 + -18 = 0
-17 + -6x + 9x2 = 0

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

Divide each side by '9'.
-1.888888889 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '1.888888889' to each side of the equation.
-1.888888889 + -0.6666666667x + 1.888888889 + x2 = 0 + 1.888888889

Reorder the terms:
-1.888888889 + 1.888888889 + -0.6666666667x + x2 = 0 + 1.888888889

Combine like terms: -1.888888889 + 1.888888889 = 0.000000000
0.000000000 + -0.6666666667x + x2 = 0 + 1.888888889
-0.6666666667x + x2 = 0 + 1.888888889

Combine like terms: 0 + 1.888888889 = 1.888888889
-0.6666666667x + x2 = 1.888888889

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 = 1.888888889 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 1.888888889 + 0.1111111112

Combine like terms: 1.888888889 + 0.1111111112 = 2.0000000002
0.1111111112 + -0.6666666667x + x2 = 2.0000000002

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 2.0000000002

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -0.3333333334 = 1.414213562 Simplifying x + -0.3333333334 = 1.414213562 Reorder the terms: -0.3333333334 + x = 1.414213562 Solving -0.3333333334 + x = 1.414213562 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 = 1.414213562 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.414213562 + 0.3333333334 x = 1.414213562 + 0.3333333334 Combine like terms: 1.414213562 + 0.3333333334 = 1.7475468954 x = 1.7475468954 Simplifying x = 1.7475468954

Subproblem 2

x + -0.3333333334 = -1.414213562 Simplifying x + -0.3333333334 = -1.414213562 Reorder the terms: -0.3333333334 + x = -1.414213562 Solving -0.3333333334 + x = -1.414213562 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 = -1.414213562 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.414213562 + 0.3333333334 x = -1.414213562 + 0.3333333334 Combine like terms: -1.414213562 + 0.3333333334 = -1.0808802286 x = -1.0808802286 Simplifying x = -1.0808802286

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.7475468954, -1.0808802286}

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