3z^2+27z-44=0

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Solution for 3z^2+27z-44=0 equation:


Simplifying
3z2 + 27z + -44 = 0

Reorder the terms:
-44 + 27z + 3z2 = 0

Solving
-44 + 27z + 3z2 = 0

Solving for variable 'z'.

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

Divide each side by '3'.
-14.66666667 + 9z + z2 = 0

Move the constant term to the right:

Add '14.66666667' to each side of the equation.
-14.66666667 + 9z + 14.66666667 + z2 = 0 + 14.66666667

Reorder the terms:
-14.66666667 + 14.66666667 + 9z + z2 = 0 + 14.66666667

Combine like terms: -14.66666667 + 14.66666667 = 0.00000000
0.00000000 + 9z + z2 = 0 + 14.66666667
9z + z2 = 0 + 14.66666667

Combine like terms: 0 + 14.66666667 = 14.66666667
9z + z2 = 14.66666667

The z term is 9z.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9z + 20.25 + z2 = 14.66666667 + 20.25

Reorder the terms:
20.25 + 9z + z2 = 14.66666667 + 20.25

Combine like terms: 14.66666667 + 20.25 = 34.91666667
20.25 + 9z + z2 = 34.91666667

Factor a perfect square on the left side:
(z + 4.5)(z + 4.5) = 34.91666667

Calculate the square root of the right side: 5.909032634

Break this problem into two subproblems by setting 
(z + 4.5) equal to 5.909032634 and -5.909032634.

Subproblem 1

z + 4.5 = 5.909032634 Simplifying z + 4.5 = 5.909032634 Reorder the terms: 4.5 + z = 5.909032634 Solving 4.5 + z = 5.909032634 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = 5.909032634 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 5.909032634 + -4.5 z = 5.909032634 + -4.5 Combine like terms: 5.909032634 + -4.5 = 1.409032634 z = 1.409032634 Simplifying z = 1.409032634

Subproblem 2

z + 4.5 = -5.909032634 Simplifying z + 4.5 = -5.909032634 Reorder the terms: 4.5 + z = -5.909032634 Solving 4.5 + z = -5.909032634 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = -5.909032634 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -5.909032634 + -4.5 z = -5.909032634 + -4.5 Combine like terms: -5.909032634 + -4.5 = -10.409032634 z = -10.409032634 Simplifying z = -10.409032634

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.409032634, -10.409032634}

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