4z^2-9z-18=0

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Solution for 4z^2-9z-18=0 equation: 


Simplifying
4z2 + -9z + -18 = 0

Reorder the terms:
-18 + -9z + 4z2 = 0

Solving
-18 + -9z + 4z2 = 0

Solving for variable 'z'.

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

Divide each side by '4'.
-4.5 + -2.25z + z2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -2.25z + 4.5 + z2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -2.25z + z2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -2.25z + z2 = 0 + 4.5
-2.25z + z2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-2.25z + z2 = 4.5

The z term is -2.25z.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25z + 1.265625 + z2 = 4.5 + 1.265625

Reorder the terms:
1.265625 + -2.25z + z2 = 4.5 + 1.265625

Combine like terms: 4.5 + 1.265625 = 5.765625
1.265625 + -2.25z + z2 = 5.765625

Factor a perfect square on the left side:
(z + -1.125)(z + -1.125) = 5.765625

Calculate the square root of the right side: 2.401171589

Break this problem into two subproblems by setting 
(z + -1.125) equal to 2.401171589 and -2.401171589.


Subproblem 1
z + -1.125 = 2.401171589

Simplifying
z + -1.125 = 2.401171589

Reorder the terms:
-1.125 + z = 2.401171589

Solving
-1.125 + z = 2.401171589

Solving for variable 'z'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + z = 2.401171589 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + z = 2.401171589 + 1.125
z = 2.401171589 + 1.125

Combine like terms: 2.401171589 + 1.125 = 3.526171589
z = 3.526171589

Simplifying
z = 3.526171589


Subproblem 2
z + -1.125 = -2.401171589

Simplifying
z + -1.125 = -2.401171589

Reorder the terms:
-1.125 + z = -2.401171589

Solving
-1.125 + z = -2.401171589

Solving for variable 'z'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + z = -2.401171589 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + z = -2.401171589 + 1.125
z = -2.401171589 + 1.125

Combine like terms: -2.401171589 + 1.125 = -1.276171589
z = -1.276171589

Simplifying
z = -1.276171589


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.526171589, -1.276171589}

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