3x(x+6)+2(x+12)=0

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Solution for 3x(x+6)+2(x+12)=0 equation:


Simplifying
3x(x + 6) + 2(x + 12) = 0

Reorder the terms:
3x(6 + x) + 2(x + 12) = 0
(6 * 3x + x * 3x) + 2(x + 12) = 0
(18x + 3x2) + 2(x + 12) = 0

Reorder the terms:
18x + 3x2 + 2(12 + x) = 0
18x + 3x2 + (12 * 2 + x * 2) = 0
18x + 3x2 + (24 + 2x) = 0

Reorder the terms:
24 + 18x + 2x + 3x2 = 0

Combine like terms: 18x + 2x = 20x
24 + 20x + 3x2 = 0

Solving
24 + 20x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
8 + 6.666666667x + x2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 6.666666667x + -8 + x2 = 0 + -8

Reorder the terms:
8 + -8 + 6.666666667x + x2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 6.666666667x + x2 = 0 + -8
6.666666667x + x2 = 0 + -8

Combine like terms: 0 + -8 = -8
6.666666667x + x2 = -8

The x term is 6.666666667x.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667x + 11.11111112 + x2 = -8 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667x + x2 = -8 + 11.11111112

Combine like terms: -8 + 11.11111112 = 3.11111112
11.11111112 + 6.666666667x + x2 = 3.11111112

Factor a perfect square on the left side:
(x + 3.333333334)(x + 3.333333334) = 3.11111112

Calculate the square root of the right side: 1.76383421

Break this problem into two subproblems by setting 
(x + 3.333333334) equal to 1.76383421 and -1.76383421.

Subproblem 1

x + 3.333333334 = 1.76383421 Simplifying x + 3.333333334 = 1.76383421 Reorder the terms: 3.333333334 + x = 1.76383421 Solving 3.333333334 + x = 1.76383421 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + x = 1.76383421 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + x = 1.76383421 + -3.333333334 x = 1.76383421 + -3.333333334 Combine like terms: 1.76383421 + -3.333333334 = -1.569499124 x = -1.569499124 Simplifying x = -1.569499124

Subproblem 2

x + 3.333333334 = -1.76383421 Simplifying x + 3.333333334 = -1.76383421 Reorder the terms: 3.333333334 + x = -1.76383421 Solving 3.333333334 + x = -1.76383421 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + x = -1.76383421 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + x = -1.76383421 + -3.333333334 x = -1.76383421 + -3.333333334 Combine like terms: -1.76383421 + -3.333333334 = -5.097167544 x = -5.097167544 Simplifying x = -5.097167544

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.569499124, -5.097167544}

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