4z^2-24z-73=0

Solution for 4z^2-24z-73=0 equation:

Simplifying
4z2 + -24z + -73 = 0

Reorder the terms:
-73 + -24z + 4z2 = 0

Solving
-73 + -24z + 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'.
-18.25 + -6z + z2 = 0

Move the constant term to the right:

Add '18.25' to each side of the equation.
-18.25 + -6z + 18.25 + z2 = 0 + 18.25

Reorder the terms:
-18.25 + 18.25 + -6z + z2 = 0 + 18.25

Combine like terms: -18.25 + 18.25 = 0.00
0.00 + -6z + z2 = 0 + 18.25
-6z + z2 = 0 + 18.25

Combine like terms: 0 + 18.25 = 18.25
-6z + z2 = 18.25

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = 18.25 + 9

Reorder the terms:
9 + -6z + z2 = 18.25 + 9

Combine like terms: 18.25 + 9 = 27.25
9 + -6z + z2 = 27.25

Factor a perfect square on the left side:
(z + -3)(z + -3) = 27.25

Calculate the square root of the right side: 5.220153254

Break this problem into two subproblems by setting 
(z + -3) equal to 5.220153254 and -5.220153254.

Subproblem 1
z + -3 = 5.220153254

Simplifying
z + -3 = 5.220153254

Reorder the terms:
-3 + z = 5.220153254

Solving
-3 + z = 5.220153254

Solving for variable 'z'.

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

Add '3' to each side of the equation.
-3 + 3 + z = 5.220153254 + 3

Combine like terms: -3 + 3 = 0
0 + z = 5.220153254 + 3
z = 5.220153254 + 3

Combine like terms: 5.220153254 + 3 = 8.220153254
z = 8.220153254

Simplifying
z = 8.220153254

Subproblem 2
z + -3 = -5.220153254

Simplifying
z + -3 = -5.220153254

Reorder the terms:
-3 + z = -5.220153254

Solving
-3 + z = -5.220153254

Solving for variable 'z'.

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

Add '3' to each side of the equation.
-3 + 3 + z = -5.220153254 + 3

Combine like terms: -3 + 3 = 0
0 + z = -5.220153254 + 3
z = -5.220153254 + 3

Combine like terms: -5.220153254 + 3 = -2.220153254
z = -2.220153254

Simplifying
z = -2.220153254

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {8.220153254, -2.220153254}