-9z^2+4z+6=0

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Solution for -9z^2+4z+6=0 equation:


Simplifying
-9z2 + 4z + 6 = 0

Reorder the terms:
6 + 4z + -9z2 = 0

Solving
6 + 4z + -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'.
-0.6666666667 + -0.4444444444z + z2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -0.4444444444z + 0.6666666667 + z2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -0.4444444444z + z2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -0.4444444444z + z2 = 0 + 0.6666666667
-0.4444444444z + z2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-0.4444444444z + z2 = 0.6666666667

The z term is -0.4444444444z.  Take half its coefficient (-0.2222222222).
Square it (0.04938271604) and add it to both sides.

Add '0.04938271604' to each side of the equation.
-0.4444444444z + 0.04938271604 + z2 = 0.6666666667 + 0.04938271604

Reorder the terms:
0.04938271604 + -0.4444444444z + z2 = 0.6666666667 + 0.04938271604

Combine like terms: 0.6666666667 + 0.04938271604 = 0.71604938274
0.04938271604 + -0.4444444444z + z2 = 0.71604938274

Factor a perfect square on the left side:
(z + -0.2222222222)(z + -0.2222222222) = 0.71604938274

Calculate the square root of the right side: 0.846197012

Break this problem into two subproblems by setting 
(z + -0.2222222222) equal to 0.846197012 and -0.846197012.

Subproblem 1

z + -0.2222222222 = 0.846197012 Simplifying z + -0.2222222222 = 0.846197012 Reorder the terms: -0.2222222222 + z = 0.846197012 Solving -0.2222222222 + z = 0.846197012 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + z = 0.846197012 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + z = 0.846197012 + 0.2222222222 z = 0.846197012 + 0.2222222222 Combine like terms: 0.846197012 + 0.2222222222 = 1.0684192342 z = 1.0684192342 Simplifying z = 1.0684192342

Subproblem 2

z + -0.2222222222 = -0.846197012 Simplifying z + -0.2222222222 = -0.846197012 Reorder the terms: -0.2222222222 + z = -0.846197012 Solving -0.2222222222 + z = -0.846197012 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + z = -0.846197012 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + z = -0.846197012 + 0.2222222222 z = -0.846197012 + 0.2222222222 Combine like terms: -0.846197012 + 0.2222222222 = -0.6239747898 z = -0.6239747898 Simplifying z = -0.6239747898

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.0684192342, -0.6239747898}

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