9z^2+12z-4=

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Solution for 9z^2+12z-4= equation:


Simplifying
9z2 + 12z + -4 = 0

Reorder the terms:
-4 + 12z + 9z2 = 0

Solving
-4 + 12z + 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'.
-0.4444444444 + 1.333333333z + z2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + 1.333333333z + 0.4444444444 + z2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + 1.333333333z + z2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + 1.333333333z + z2 = 0 + 0.4444444444
1.333333333z + z2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
1.333333333z + z2 = 0.4444444444

The z term is 1.333333333z.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333z + 0.4444444442 + z2 = 0.4444444444 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333z + z2 = 0.4444444444 + 0.4444444442

Combine like terms: 0.4444444444 + 0.4444444442 = 0.8888888886
0.4444444442 + 1.333333333z + z2 = 0.8888888886

Factor a perfect square on the left side:
(z + 0.6666666665)(z + 0.6666666665) = 0.8888888886

Calculate the square root of the right side: 0.942809041

Break this problem into two subproblems by setting 
(z + 0.6666666665) equal to 0.942809041 and -0.942809041.

Subproblem 1

z + 0.6666666665 = 0.942809041 Simplifying z + 0.6666666665 = 0.942809041 Reorder the terms: 0.6666666665 + z = 0.942809041 Solving 0.6666666665 + z = 0.942809041 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + z = 0.942809041 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + z = 0.942809041 + -0.6666666665 z = 0.942809041 + -0.6666666665 Combine like terms: 0.942809041 + -0.6666666665 = 0.2761423745 z = 0.2761423745 Simplifying z = 0.2761423745

Subproblem 2

z + 0.6666666665 = -0.942809041 Simplifying z + 0.6666666665 = -0.942809041 Reorder the terms: 0.6666666665 + z = -0.942809041 Solving 0.6666666665 + z = -0.942809041 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + z = -0.942809041 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + z = -0.942809041 + -0.6666666665 z = -0.942809041 + -0.6666666665 Combine like terms: -0.942809041 + -0.6666666665 = -1.6094757075 z = -1.6094757075 Simplifying z = -1.6094757075

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.2761423745, -1.6094757075}

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