9q^2+6q-1=0

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Solution for 9q^2+6q-1=0 equation: 


Simplifying
9q2 + 6q + -1 = 0

Reorder the terms:
-1 + 6q + 9q2 = 0

Solving
-1 + 6q + 9q2 = 0

Solving for variable 'q'.

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

Divide each side by '9'.
-0.1111111111 + 0.6666666667q + q2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 0.6666666667q + 0.1111111111 + q2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + 0.6666666667q + q2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + 0.6666666667q + q2 = 0 + 0.1111111111
0.6666666667q + q2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
0.6666666667q + q2 = 0.1111111111

The q term is 0.6666666667q.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667q + 0.1111111112 + q2 = 0.1111111111 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667q + q2 = 0.1111111111 + 0.1111111112

Combine like terms: 0.1111111111 + 0.1111111112 = 0.2222222223
0.1111111112 + 0.6666666667q + q2 = 0.2222222223

Factor a perfect square on the left side:
(q + 0.3333333334)(q + 0.3333333334) = 0.2222222223

Calculate the square root of the right side: 0.471404521

Break this problem into two subproblems by setting 
(q + 0.3333333334) equal to 0.471404521 and -0.471404521.


Subproblem 1
q + 0.3333333334 = 0.471404521

Simplifying
q + 0.3333333334 = 0.471404521

Reorder the terms:
0.3333333334 + q = 0.471404521

Solving
0.3333333334 + q = 0.471404521

Solving for variable 'q'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + q = 0.471404521 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + q = 0.471404521 + -0.3333333334
q = 0.471404521 + -0.3333333334

Combine like terms: 0.471404521 + -0.3333333334 = 0.1380711876
q = 0.1380711876

Simplifying
q = 0.1380711876


Subproblem 2
q + 0.3333333334 = -0.471404521

Simplifying
q + 0.3333333334 = -0.471404521

Reorder the terms:
0.3333333334 + q = -0.471404521

Solving
0.3333333334 + q = -0.471404521

Solving for variable 'q'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + q = -0.471404521 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + q = -0.471404521 + -0.3333333334
q = -0.471404521 + -0.3333333334

Combine like terms: -0.471404521 + -0.3333333334 = -0.8047378544
q = -0.8047378544

Simplifying
q = -0.8047378544


Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {0.1380711876, -0.8047378544}

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