3q^2-q-9=0

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Solution for 3q^2-q-9=0 equation: 


Simplifying
3q2 + -1q + -9 = 0

Reorder the terms:
-9 + -1q + 3q2 = 0

Solving
-9 + -1q + 3q2 = 0

Solving for variable 'q'.

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

Divide each side by '3'.
-3 + -0.3333333333q + q2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.3333333333q + 3 + q2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.3333333333q + q2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.3333333333q + q2 = 0 + 3
-0.3333333333q + q2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.3333333333q + q2 = 3

The q term is -0.3333333333q.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333q + 0.02777777779 + q2 = 3 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333q + q2 = 3 + 0.02777777779

Combine like terms: 3 + 0.02777777779 = 3.02777777779
0.02777777779 + -0.3333333333q + q2 = 3.02777777779

Factor a perfect square on the left side:
(q + -0.1666666667)(q + -0.1666666667) = 3.02777777779

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(q + -0.1666666667) equal to 1.740051085 and -1.740051085.


Subproblem 1
q + -0.1666666667 = 1.740051085

Simplifying
q + -0.1666666667 = 1.740051085

Reorder the terms:
-0.1666666667 + q = 1.740051085

Solving
-0.1666666667 + q = 1.740051085

Solving for variable 'q'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + q = 1.740051085 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + q = 1.740051085 + 0.1666666667
q = 1.740051085 + 0.1666666667

Combine like terms: 1.740051085 + 0.1666666667 = 1.9067177517
q = 1.9067177517

Simplifying
q = 1.9067177517


Subproblem 2
q + -0.1666666667 = -1.740051085

Simplifying
q + -0.1666666667 = -1.740051085

Reorder the terms:
-0.1666666667 + q = -1.740051085

Solving
-0.1666666667 + q = -1.740051085

Solving for variable 'q'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + q = -1.740051085 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + q = -1.740051085 + 0.1666666667
q = -1.740051085 + 0.1666666667

Combine like terms: -1.740051085 + 0.1666666667 = -1.5733844183
q = -1.5733844183

Simplifying
q = -1.5733844183


Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {1.9067177517, -1.5733844183}

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