(12x^2+46x-174)=0

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Solution for (12x^2+46x-174)=0 equation: 


Simplifying
(12x2 + 46x + -174) = 0

Reorder the terms:
(-174 + 46x + 12x2) = 0

Remove parenthesis around (-174 + 46x + 12x2)
-174 + 46x + 12x2 = 0

Solving
-174 + 46x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-87 + 23x + 6x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-87 + 23x + 6x2)' equal to zero and attempt to solve:

Simplifying
-87 + 23x + 6x2 = 0

Solving
-87 + 23x + 6x2 = 0

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

Divide each side by '6'.
-14.5 + 3.833333333x + x2 = 0

Move the constant term to the right:

Add '14.5' to each side of the equation.
-14.5 + 3.833333333x + 14.5 + x2 = 0 + 14.5

Reorder the terms:
-14.5 + 14.5 + 3.833333333x + x2 = 0 + 14.5

Combine like terms: -14.5 + 14.5 = 0.0
0.0 + 3.833333333x + x2 = 0 + 14.5
3.833333333x + x2 = 0 + 14.5

Combine like terms: 0 + 14.5 = 14.5
3.833333333x + x2 = 14.5

The x term is 3.833333333x.  Take half its coefficient (1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
3.833333333x + 3.673611112 + x2 = 14.5 + 3.673611112

Reorder the terms:
3.673611112 + 3.833333333x + x2 = 14.5 + 3.673611112

Combine like terms: 14.5 + 3.673611112 = 18.173611112
3.673611112 + 3.833333333x + x2 = 18.173611112

Factor a perfect square on the left side:
(x + 1.916666667)(x + 1.916666667) = 18.173611112

Calculate the square root of the right side: 4.263051854

Break this problem into two subproblems by setting 
(x + 1.916666667) equal to 4.263051854 and -4.263051854.


Subproblem 1
x + 1.916666667 = 4.263051854

Simplifying
x + 1.916666667 = 4.263051854

Reorder the terms:
1.916666667 + x = 4.263051854

Solving
1.916666667 + x = 4.263051854

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.916666667' to each side of the equation.
1.916666667 + -1.916666667 + x = 4.263051854 + -1.916666667

Combine like terms: 1.916666667 + -1.916666667 = 0.000000000
0.000000000 + x = 4.263051854 + -1.916666667
x = 4.263051854 + -1.916666667

Combine like terms: 4.263051854 + -1.916666667 = 2.346385187
x = 2.346385187

Simplifying
x = 2.346385187


Subproblem 2
x + 1.916666667 = -4.263051854

Simplifying
x + 1.916666667 = -4.263051854

Reorder the terms:
1.916666667 + x = -4.263051854

Solving
1.916666667 + x = -4.263051854

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.916666667' to each side of the equation.
1.916666667 + -1.916666667 + x = -4.263051854 + -1.916666667

Combine like terms: 1.916666667 + -1.916666667 = 0.000000000
0.000000000 + x = -4.263051854 + -1.916666667
x = -4.263051854 + -1.916666667

Combine like terms: -4.263051854 + -1.916666667 = -6.179718521
x = -6.179718521

Simplifying
x = -6.179718521


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.346385187, -6.179718521}
Solution
x = {2.346385187, -6.179718521}

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