3x(x-1.8)=2x+1

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Solution for 3x(x-1.8)=2x+1 equation:


Simplifying
3x(x + -1.8) = 2x + 1

Reorder the terms:
3x(-1.8 + x) = 2x + 1
(-1.8 * 3x + x * 3x) = 2x + 1
(-5.4x + 3x2) = 2x + 1

Reorder the terms:
-5.4x + 3x2 = 1 + 2x

Solving
-5.4x + 3x2 = 1 + 2x

Solving for variable 'x'.

Reorder the terms:
-1 + -5.4x + -2x + 3x2 = 1 + 2x + -1 + -2x

Combine like terms: -5.4x + -2x = -7.4x
-1 + -7.4x + 3x2 = 1 + 2x + -1 + -2x

Reorder the terms:
-1 + -7.4x + 3x2 = 1 + -1 + 2x + -2x

Combine like terms: 1 + -1 = 0
-1 + -7.4x + 3x2 = 0 + 2x + -2x
-1 + -7.4x + 3x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-1 + -7.4x + 3x2 = 0

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

Divide each side by '3'.
-0.3333333333 + -2.466666667x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -2.466666667x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -2.466666667x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -2.466666667x + x2 = 0 + 0.3333333333
-2.466666667x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-2.466666667x + x2 = 0.3333333333

The x term is -2.466666667x.  Take half its coefficient (-1.233333334).
Square it (1.521111113) and add it to both sides.

Add '1.521111113' to each side of the equation.
-2.466666667x + 1.521111113 + x2 = 0.3333333333 + 1.521111113

Reorder the terms:
1.521111113 + -2.466666667x + x2 = 0.3333333333 + 1.521111113

Combine like terms: 0.3333333333 + 1.521111113 = 1.8544444463
1.521111113 + -2.466666667x + x2 = 1.8544444463

Factor a perfect square on the left side:
(x + -1.233333334)(x + -1.233333334) = 1.8544444463

Calculate the square root of the right side: 1.361779882

Break this problem into two subproblems by setting 
(x + -1.233333334) equal to 1.361779882 and -1.361779882.

Subproblem 1

x + -1.233333334 = 1.361779882 Simplifying x + -1.233333334 = 1.361779882 Reorder the terms: -1.233333334 + x = 1.361779882 Solving -1.233333334 + x = 1.361779882 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.233333334' to each side of the equation. -1.233333334 + 1.233333334 + x = 1.361779882 + 1.233333334 Combine like terms: -1.233333334 + 1.233333334 = 0.000000000 0.000000000 + x = 1.361779882 + 1.233333334 x = 1.361779882 + 1.233333334 Combine like terms: 1.361779882 + 1.233333334 = 2.595113216 x = 2.595113216 Simplifying x = 2.595113216

Subproblem 2

x + -1.233333334 = -1.361779882 Simplifying x + -1.233333334 = -1.361779882 Reorder the terms: -1.233333334 + x = -1.361779882 Solving -1.233333334 + x = -1.361779882 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.233333334' to each side of the equation. -1.233333334 + 1.233333334 + x = -1.361779882 + 1.233333334 Combine like terms: -1.233333334 + 1.233333334 = 0.000000000 0.000000000 + x = -1.361779882 + 1.233333334 x = -1.361779882 + 1.233333334 Combine like terms: -1.361779882 + 1.233333334 = -0.128446548 x = -0.128446548 Simplifying x = -0.128446548

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.595113216, -0.128446548}

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