9z^2-23z-116=0

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Solution for 9z^2-23z-116=0 equation:


Simplifying
9z2 + -23z + -116 = 0

Reorder the terms:
-116 + -23z + 9z2 = 0

Solving
-116 + -23z + 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'.
-12.88888889 + -2.555555556z + z2 = 0

Move the constant term to the right:

Add '12.88888889' to each side of the equation.
-12.88888889 + -2.555555556z + 12.88888889 + z2 = 0 + 12.88888889

Reorder the terms:
-12.88888889 + 12.88888889 + -2.555555556z + z2 = 0 + 12.88888889

Combine like terms: -12.88888889 + 12.88888889 = 0.00000000
0.00000000 + -2.555555556z + z2 = 0 + 12.88888889
-2.555555556z + z2 = 0 + 12.88888889

Combine like terms: 0 + 12.88888889 = 12.88888889
-2.555555556z + z2 = 12.88888889

The z term is -2.555555556z.  Take half its coefficient (-1.277777778).
Square it (1.632716050) and add it to both sides.

Add '1.632716050' to each side of the equation.
-2.555555556z + 1.632716050 + z2 = 12.88888889 + 1.632716050

Reorder the terms:
1.632716050 + -2.555555556z + z2 = 12.88888889 + 1.632716050

Combine like terms: 12.88888889 + 1.632716050 = 14.52160494
1.632716050 + -2.555555556z + z2 = 14.52160494

Factor a perfect square on the left side:
(z + -1.277777778)(z + -1.277777778) = 14.52160494

Calculate the square root of the right side: 3.810722365

Break this problem into two subproblems by setting 
(z + -1.277777778) equal to 3.810722365 and -3.810722365.

Subproblem 1

z + -1.277777778 = 3.810722365 Simplifying z + -1.277777778 = 3.810722365 Reorder the terms: -1.277777778 + z = 3.810722365 Solving -1.277777778 + z = 3.810722365 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.277777778' to each side of the equation. -1.277777778 + 1.277777778 + z = 3.810722365 + 1.277777778 Combine like terms: -1.277777778 + 1.277777778 = 0.000000000 0.000000000 + z = 3.810722365 + 1.277777778 z = 3.810722365 + 1.277777778 Combine like terms: 3.810722365 + 1.277777778 = 5.088500143 z = 5.088500143 Simplifying z = 5.088500143

Subproblem 2

z + -1.277777778 = -3.810722365 Simplifying z + -1.277777778 = -3.810722365 Reorder the terms: -1.277777778 + z = -3.810722365 Solving -1.277777778 + z = -3.810722365 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.277777778' to each side of the equation. -1.277777778 + 1.277777778 + z = -3.810722365 + 1.277777778 Combine like terms: -1.277777778 + 1.277777778 = 0.000000000 0.000000000 + z = -3.810722365 + 1.277777778 z = -3.810722365 + 1.277777778 Combine like terms: -3.810722365 + 1.277777778 = -2.532944587 z = -2.532944587 Simplifying z = -2.532944587

Solution

The solution to the problem is based on the solutions from the subproblems. z = {5.088500143, -2.532944587}

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