-6x^2-30x-23=0

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Solution for -6x^2-30x-23=0 equation: 


Simplifying
-6x2 + -30x + -23 = 0

Reorder the terms:
-23 + -30x + -6x2 = 0

Solving
-23 + -30x + -6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '-1'.
-1(23 + 30x + 6x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(23 + 30x + 6x2)' equal to zero and attempt to solve:

Simplifying
23 + 30x + 6x2 = 0

Solving
23 + 30x + 6x2 = 0

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

Divide each side by '6'.
3.833333333 + 5x + x2 = 0

Move the constant term to the right:

Add '-3.833333333' to each side of the equation.
3.833333333 + 5x + -3.833333333 + x2 = 0 + -3.833333333

Reorder the terms:
3.833333333 + -3.833333333 + 5x + x2 = 0 + -3.833333333

Combine like terms: 3.833333333 + -3.833333333 = 0.000000000
0.000000000 + 5x + x2 = 0 + -3.833333333
5x + x2 = 0 + -3.833333333

Combine like terms: 0 + -3.833333333 = -3.833333333
5x + x2 = -3.833333333

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = -3.833333333 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -3.833333333 + 6.25

Combine like terms: -3.833333333 + 6.25 = 2.416666667
6.25 + 5x + x2 = 2.416666667

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 2.416666667

Calculate the square root of the right side: 1.554563176

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1.554563176 and -1.554563176.


Subproblem 1
x + 2.5 = 1.554563176

Simplifying
x + 2.5 = 1.554563176

Reorder the terms:
2.5 + x = 1.554563176

Solving
2.5 + x = 1.554563176

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 1.554563176 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 1.554563176 + -2.5
x = 1.554563176 + -2.5

Combine like terms: 1.554563176 + -2.5 = -0.945436824
x = -0.945436824

Simplifying
x = -0.945436824


Subproblem 2
x + 2.5 = -1.554563176

Simplifying
x + 2.5 = -1.554563176

Reorder the terms:
2.5 + x = -1.554563176

Solving
2.5 + x = -1.554563176

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -1.554563176 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -1.554563176 + -2.5
x = -1.554563176 + -2.5

Combine like terms: -1.554563176 + -2.5 = -4.054563176
x = -4.054563176

Simplifying
x = -4.054563176


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.945436824, -4.054563176}
Solution
x = {-0.945436824, -4.054563176}

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