2x(x+3)+3=0

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


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

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

Reorder the terms:
3 + 6x + 2x2 = 0

Solving
3 + 6x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
1.5 + 3x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 3x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 3x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 3x + x2 = 0 + -1.5
3x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
3x + x2 = -1.5

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 = -1.5 + 2.25

Reorder the terms:
2.25 + 3x + x2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + 3x + x2 = 0.75

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + 1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

x + 1.5 = 0.866025404 Simplifying x + 1.5 = 0.866025404 Reorder the terms: 1.5 + x = 0.866025404 Solving 1.5 + x = 0.866025404 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 = 0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 0.866025404 + -1.5 x = 0.866025404 + -1.5 Combine like terms: 0.866025404 + -1.5 = -0.633974596 x = -0.633974596 Simplifying x = -0.633974596

Subproblem 2

x + 1.5 = -0.866025404 Simplifying x + 1.5 = -0.866025404 Reorder the terms: 1.5 + x = -0.866025404 Solving 1.5 + x = -0.866025404 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 = -0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -0.866025404 + -1.5 x = -0.866025404 + -1.5 Combine like terms: -0.866025404 + -1.5 = -2.366025404 x = -2.366025404 Simplifying x = -2.366025404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.633974596, -2.366025404}

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