0=3x(x+2)-5

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Solution for 0=3x(x+2)-5 equation:


Simplifying
0 = 3x(x + 2) + -5

Reorder the terms:
0 = 3x(2 + x) + -5
0 = (2 * 3x + x * 3x) + -5
0 = (6x + 3x2) + -5

Reorder the terms:
0 = -5 + 6x + 3x2

Solving
0 = -5 + 6x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + 5 = 5
5 + -6x + -3x2 = -5 + 6x + 3x2 + 5 + -6x + -3x2

Reorder the terms:
5 + -6x + -3x2 = -5 + 5 + 6x + -6x + 3x2 + -3x2

Combine like terms: -5 + 5 = 0
5 + -6x + -3x2 = 0 + 6x + -6x + 3x2 + -3x2
5 + -6x + -3x2 = 6x + -6x + 3x2 + -3x2

Combine like terms: 6x + -6x = 0
5 + -6x + -3x2 = 0 + 3x2 + -3x2
5 + -6x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
5 + -6x + -3x2 = 0

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

Divide each side by '-3'.
-1.666666667 + 2x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 2x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 2x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 2x + x2 = 0 + 1.666666667
2x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
2x + x2 = 1.666666667

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 1.666666667 + 1

Reorder the terms:
1 + 2x + x2 = 1.666666667 + 1

Combine like terms: 1.666666667 + 1 = 2.666666667
1 + 2x + x2 = 2.666666667

Factor a perfect square on the left side:
(x + 1)(x + 1) = 2.666666667

Calculate the square root of the right side: 1.632993162

Break this problem into two subproblems by setting 
(x + 1) equal to 1.632993162 and -1.632993162.

Subproblem 1

x + 1 = 1.632993162 Simplifying x + 1 = 1.632993162 Reorder the terms: 1 + x = 1.632993162 Solving 1 + x = 1.632993162 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 1.632993162 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.632993162 + -1 x = 1.632993162 + -1 Combine like terms: 1.632993162 + -1 = 0.632993162 x = 0.632993162 Simplifying x = 0.632993162

Subproblem 2

x + 1 = -1.632993162 Simplifying x + 1 = -1.632993162 Reorder the terms: 1 + x = -1.632993162 Solving 1 + x = -1.632993162 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -1.632993162 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.632993162 + -1 x = -1.632993162 + -1 Combine like terms: -1.632993162 + -1 = -2.632993162 x = -2.632993162 Simplifying x = -2.632993162

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.632993162, -2.632993162}

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