3x(x-3)=1

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


Simplifying
3x(x + -3) = 1

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

Solving
-9x + 3x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -9x + 3x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -9x + 3x2 = 0

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

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

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -3x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -3x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -3x + x2 = 0 + 0.3333333333
-3x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-3x + x2 = 0.3333333333

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 0.3333333333 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 0.3333333333 + 2.25

Combine like terms: 0.3333333333 + 2.25 = 2.5833333333
2.25 + -3x + x2 = 2.5833333333

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 2.5833333333

Calculate the square root of the right side: 1.607275127

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.607275127 and -1.607275127.

Subproblem 1

x + -1.5 = 1.607275127 Simplifying x + -1.5 = 1.607275127 Reorder the terms: -1.5 + x = 1.607275127 Solving -1.5 + x = 1.607275127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.607275127 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.607275127 + 1.5 x = 1.607275127 + 1.5 Combine like terms: 1.607275127 + 1.5 = 3.107275127 x = 3.107275127 Simplifying x = 3.107275127

Subproblem 2

x + -1.5 = -1.607275127 Simplifying x + -1.5 = -1.607275127 Reorder the terms: -1.5 + x = -1.607275127 Solving -1.5 + x = -1.607275127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.607275127 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.607275127 + 1.5 x = -1.607275127 + 1.5 Combine like terms: -1.607275127 + 1.5 = -0.107275127 x = -0.107275127 Simplifying x = -0.107275127

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.107275127, -0.107275127}

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