z(5-z)-8(3-4z)=16

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Solution for z(5-z)-8(3-4z)=16 equation: 


Simplifying
z(5 + -1z) + -8(3 + -4z) = 16
(5 * z + -1z * z) + -8(3 + -4z) = 16
(5z + -1z2) + -8(3 + -4z) = 16
5z + -1z2 + (3 * -8 + -4z * -8) = 16
5z + -1z2 + (-24 + 32z) = 16

Reorder the terms:
-24 + 5z + 32z + -1z2 = 16

Combine like terms: 5z + 32z = 37z
-24 + 37z + -1z2 = 16

Solving
-24 + 37z + -1z2 = 16

Solving for variable 'z'.

Reorder the terms:
-24 + -16 + 37z + -1z2 = 16 + -16

Combine like terms: -24 + -16 = -40
-40 + 37z + -1z2 = 16 + -16

Combine like terms: 16 + -16 = 0
-40 + 37z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
40 + -37z + z2 = 0

Move the constant term to the right:

Add '-40' to each side of the equation.
40 + -37z + -40 + z2 = 0 + -40

Reorder the terms:
40 + -40 + -37z + z2 = 0 + -40

Combine like terms: 40 + -40 = 0
0 + -37z + z2 = 0 + -40
-37z + z2 = 0 + -40

Combine like terms: 0 + -40 = -40
-37z + z2 = -40

The z term is -37z.  Take half its coefficient (-18.5).
Square it (342.25) and add it to both sides.

Add '342.25' to each side of the equation.
-37z + 342.25 + z2 = -40 + 342.25

Reorder the terms:
342.25 + -37z + z2 = -40 + 342.25

Combine like terms: -40 + 342.25 = 302.25
342.25 + -37z + z2 = 302.25

Factor a perfect square on the left side:
(z + -18.5)(z + -18.5) = 302.25

Calculate the square root of the right side: 17.385338651

Break this problem into two subproblems by setting 
(z + -18.5) equal to 17.385338651 and -17.385338651.


Subproblem 1
z + -18.5 = 17.385338651

Simplifying
z + -18.5 = 17.385338651

Reorder the terms:
-18.5 + z = 17.385338651

Solving
-18.5 + z = 17.385338651

Solving for variable 'z'.

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

Add '18.5' to each side of the equation.
-18.5 + 18.5 + z = 17.385338651 + 18.5

Combine like terms: -18.5 + 18.5 = 0.0
0.0 + z = 17.385338651 + 18.5
z = 17.385338651 + 18.5

Combine like terms: 17.385338651 + 18.5 = 35.885338651
z = 35.885338651

Simplifying
z = 35.885338651


Subproblem 2
z + -18.5 = -17.385338651

Simplifying
z + -18.5 = -17.385338651

Reorder the terms:
-18.5 + z = -17.385338651

Solving
-18.5 + z = -17.385338651

Solving for variable 'z'.

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

Add '18.5' to each side of the equation.
-18.5 + 18.5 + z = -17.385338651 + 18.5

Combine like terms: -18.5 + 18.5 = 0.0
0.0 + z = -17.385338651 + 18.5
z = -17.385338651 + 18.5

Combine like terms: -17.385338651 + 18.5 = 1.114661349
z = 1.114661349

Simplifying
z = 1.114661349


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {35.885338651, 1.114661349}

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