(8-z)(3z)-7=0

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Solution for (8-z)(3z)-7=0 equation: 


Simplifying
(8 + -1z)(3z) + -7 = 0

Remove parenthesis around (3z)
(8 + -1z) * 3z + -7 = 0

Reorder the terms for easier multiplication:
3z(8 + -1z) + -7 = 0
(8 * 3z + -1z * 3z) + -7 = 0
(24z + -3z2) + -7 = 0

Reorder the terms:
-7 + 24z + -3z2 = 0

Solving
-7 + 24z + -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'.
2.333333333 + -8z + z2 = 0

Move the constant term to the right:

Add '-2.333333333' to each side of the equation.
2.333333333 + -8z + -2.333333333 + z2 = 0 + -2.333333333

Reorder the terms:
2.333333333 + -2.333333333 + -8z + z2 = 0 + -2.333333333

Combine like terms: 2.333333333 + -2.333333333 = 0.000000000
0.000000000 + -8z + z2 = 0 + -2.333333333
-8z + z2 = 0 + -2.333333333

Combine like terms: 0 + -2.333333333 = -2.333333333
-8z + z2 = -2.333333333

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = -2.333333333 + 16

Reorder the terms:
16 + -8z + z2 = -2.333333333 + 16

Combine like terms: -2.333333333 + 16 = 13.666666667
16 + -8z + z2 = 13.666666667

Factor a perfect square on the left side:
(z + -4)(z + -4) = 13.666666667

Calculate the square root of the right side: 3.696845502

Break this problem into two subproblems by setting 
(z + -4) equal to 3.696845502 and -3.696845502.


Subproblem 1
z + -4 = 3.696845502

Simplifying
z + -4 = 3.696845502

Reorder the terms:
-4 + z = 3.696845502

Solving
-4 + z = 3.696845502

Solving for variable 'z'.

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

Add '4' to each side of the equation.
-4 + 4 + z = 3.696845502 + 4

Combine like terms: -4 + 4 = 0
0 + z = 3.696845502 + 4
z = 3.696845502 + 4

Combine like terms: 3.696845502 + 4 = 7.696845502
z = 7.696845502

Simplifying
z = 7.696845502


Subproblem 2
z + -4 = -3.696845502

Simplifying
z + -4 = -3.696845502

Reorder the terms:
-4 + z = -3.696845502

Solving
-4 + z = -3.696845502

Solving for variable 'z'.

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

Add '4' to each side of the equation.
-4 + 4 + z = -3.696845502 + 4

Combine like terms: -4 + 4 = 0
0 + z = -3.696845502 + 4
z = -3.696845502 + 4

Combine like terms: -3.696845502 + 4 = 0.303154498
z = 0.303154498

Simplifying
z = 0.303154498


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {7.696845502, 0.303154498}

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