-4z(z-7)-(5-6z)=z

Solution for -4z(z-7)-(5-6z)=z equation:

Simplifying
-4z(z + -7) + -1(5 + -6z) = z

Reorder the terms:
-4z(-7 + z) + -1(5 + -6z) = z
(-7 * -4z + z * -4z) + -1(5 + -6z) = z
(28z + -4z2) + -1(5 + -6z) = z
28z + -4z2 + (5 * -1 + -6z * -1) = z
28z + -4z2 + (-5 + 6z) = z

Reorder the terms:
-5 + 28z + 6z + -4z2 = z

Combine like terms: 28z + 6z = 34z
-5 + 34z + -4z2 = z

Solving
-5 + 34z + -4z2 = z

Solving for variable 'z'.

Reorder the terms:
-5 + 34z + -1z + -4z2 = z + -1z

Combine like terms: 34z + -1z = 33z
-5 + 33z + -4z2 = z + -1z

Combine like terms: z + -1z = 0
-5 + 33z + -4z2 = 0

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

Divide each side by '-4'.
1.25 + -8.25z + z2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + -8.25z + -1.25 + z2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + -8.25z + z2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + -8.25z + z2 = 0 + -1.25
-8.25z + z2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
-8.25z + z2 = -1.25

The z term is -8.25z.  Take half its coefficient (-4.125).
Square it (17.015625) and add it to both sides.

Add '17.015625' to each side of the equation.
-8.25z + 17.015625 + z2 = -1.25 + 17.015625

Reorder the terms:
17.015625 + -8.25z + z2 = -1.25 + 17.015625

Combine like terms: -1.25 + 17.015625 = 15.765625
17.015625 + -8.25z + z2 = 15.765625

Factor a perfect square on the left side:
(z + -4.125)(z + -4.125) = 15.765625

Calculate the square root of the right side: 3.970595044

Break this problem into two subproblems by setting 
(z + -4.125) equal to 3.970595044 and -3.970595044.

Subproblem 1
z + -4.125 = 3.970595044

Simplifying
z + -4.125 = 3.970595044

Reorder the terms:
-4.125 + z = 3.970595044

Solving
-4.125 + z = 3.970595044

Solving for variable 'z'.

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

Add '4.125' to each side of the equation.
-4.125 + 4.125 + z = 3.970595044 + 4.125

Combine like terms: -4.125 + 4.125 = 0.000
0.000 + z = 3.970595044 + 4.125
z = 3.970595044 + 4.125

Combine like terms: 3.970595044 + 4.125 = 8.095595044
z = 8.095595044

Simplifying
z = 8.095595044

Subproblem 2
z + -4.125 = -3.970595044

Simplifying
z + -4.125 = -3.970595044

Reorder the terms:
-4.125 + z = -3.970595044

Solving
-4.125 + z = -3.970595044

Solving for variable 'z'.

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

Add '4.125' to each side of the equation.
-4.125 + 4.125 + z = -3.970595044 + 4.125

Combine like terms: -4.125 + 4.125 = 0.000
0.000 + z = -3.970595044 + 4.125
z = -3.970595044 + 4.125

Combine like terms: -3.970595044 + 4.125 = 0.154404956
z = 0.154404956

Simplifying
z = 0.154404956

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {8.095595044, 0.154404956}