9z^2+6z-2=0

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


Simplifying
9z2 + 6z + -2 = 0

Reorder the terms:
-2 + 6z + 9z2 = 0

Solving
-2 + 6z + 9z2 = 0

Solving for variable 'z'.

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

Divide each side by '9'.
-0.2222222222 + 0.6666666667z + z2 = 0

Move the constant term to the right:

Add '0.2222222222' to each side of the equation.
-0.2222222222 + 0.6666666667z + 0.2222222222 + z2 = 0 + 0.2222222222

Reorder the terms:
-0.2222222222 + 0.2222222222 + 0.6666666667z + z2 = 0 + 0.2222222222

Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000
0.0000000000 + 0.6666666667z + z2 = 0 + 0.2222222222
0.6666666667z + z2 = 0 + 0.2222222222

Combine like terms: 0 + 0.2222222222 = 0.2222222222
0.6666666667z + z2 = 0.2222222222

The z term is 0.6666666667z.  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.
0.6666666667z + 0.1111111112 + z2 = 0.2222222222 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667z + z2 = 0.2222222222 + 0.1111111112

Combine like terms: 0.2222222222 + 0.1111111112 = 0.3333333334
0.1111111112 + 0.6666666667z + z2 = 0.3333333334

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

Calculate the square root of the right side: 0.577350269

Break this problem into two subproblems by setting 
(z + 0.3333333334) equal to 0.577350269 and -0.577350269.


Subproblem 1
z + 0.3333333334 = 0.577350269

Simplifying
z + 0.3333333334 = 0.577350269

Reorder the terms:
0.3333333334 + z = 0.577350269

Solving
0.3333333334 + z = 0.577350269

Solving for variable 'z'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + z = 0.577350269 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + z = 0.577350269 + -0.3333333334
z = 0.577350269 + -0.3333333334

Combine like terms: 0.577350269 + -0.3333333334 = 0.2440169356
z = 0.2440169356

Simplifying
z = 0.2440169356


Subproblem 2
z + 0.3333333334 = -0.577350269

Simplifying
z + 0.3333333334 = -0.577350269

Reorder the terms:
0.3333333334 + z = -0.577350269

Solving
0.3333333334 + z = -0.577350269

Solving for variable 'z'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + z = -0.577350269 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + z = -0.577350269 + -0.3333333334
z = -0.577350269 + -0.3333333334

Combine like terms: -0.577350269 + -0.3333333334 = -0.9106836024
z = -0.9106836024

Simplifying
z = -0.9106836024


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.2440169356, -0.9106836024}

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