12q^2-19q-5=0

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Solution for 12q^2-19q-5=0 equation: 


Simplifying
12q2 + -19q + -5 = 0

Reorder the terms:
-5 + -19q + 12q2 = 0

Solving
-5 + -19q + 12q2 = 0

Solving for variable 'q'.

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

Divide each side by '12'.
-0.4166666667 + -1.583333333q + q2 = 0

Move the constant term to the right:

Add '0.4166666667' to each side of the equation.
-0.4166666667 + -1.583333333q + 0.4166666667 + q2 = 0 + 0.4166666667

Reorder the terms:
-0.4166666667 + 0.4166666667 + -1.583333333q + q2 = 0 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + -1.583333333q + q2 = 0 + 0.4166666667
-1.583333333q + q2 = 0 + 0.4166666667

Combine like terms: 0 + 0.4166666667 = 0.4166666667
-1.583333333q + q2 = 0.4166666667

The q term is -1.583333333q.  Take half its coefficient (-0.7916666665).
Square it (0.6267361108) and add it to both sides.

Add '0.6267361108' to each side of the equation.
-1.583333333q + 0.6267361108 + q2 = 0.4166666667 + 0.6267361108

Reorder the terms:
0.6267361108 + -1.583333333q + q2 = 0.4166666667 + 0.6267361108

Combine like terms: 0.4166666667 + 0.6267361108 = 1.0434027775
0.6267361108 + -1.583333333q + q2 = 1.0434027775

Factor a perfect square on the left side:
(q + -0.7916666665)(q + -0.7916666665) = 1.0434027775

Calculate the square root of the right side: 1.021470889

Break this problem into two subproblems by setting 
(q + -0.7916666665) equal to 1.021470889 and -1.021470889.


Subproblem 1
q + -0.7916666665 = 1.021470889

Simplifying
q + -0.7916666665 = 1.021470889

Reorder the terms:
-0.7916666665 + q = 1.021470889

Solving
-0.7916666665 + q = 1.021470889

Solving for variable 'q'.

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

Add '0.7916666665' to each side of the equation.
-0.7916666665 + 0.7916666665 + q = 1.021470889 + 0.7916666665

Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000
0.0000000000 + q = 1.021470889 + 0.7916666665
q = 1.021470889 + 0.7916666665

Combine like terms: 1.021470889 + 0.7916666665 = 1.8131375555
q = 1.8131375555

Simplifying
q = 1.8131375555


Subproblem 2
q + -0.7916666665 = -1.021470889

Simplifying
q + -0.7916666665 = -1.021470889

Reorder the terms:
-0.7916666665 + q = -1.021470889

Solving
-0.7916666665 + q = -1.021470889

Solving for variable 'q'.

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

Add '0.7916666665' to each side of the equation.
-0.7916666665 + 0.7916666665 + q = -1.021470889 + 0.7916666665

Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000
0.0000000000 + q = -1.021470889 + 0.7916666665
q = -1.021470889 + 0.7916666665

Combine like terms: -1.021470889 + 0.7916666665 = -0.2298042225
q = -0.2298042225

Simplifying
q = -0.2298042225


Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {1.8131375555, -0.2298042225}

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