5z^2-10z-5=-12z

Solution for 5z^2-10z-5=-12z equation:

Simplifying
5z2 + -10z + -5 = -12z

Reorder the terms:
-5 + -10z + 5z2 = -12z

Solving
-5 + -10z + 5z2 = -12z

Solving for variable 'z'.

Reorder the terms:
-5 + -10z + 12z + 5z2 = -12z + 12z

Combine like terms: -10z + 12z = 2z
-5 + 2z + 5z2 = -12z + 12z

Combine like terms: -12z + 12z = 0
-5 + 2z + 5z2 = 0

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

Divide each side by '5'.
-1 + 0.4z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.4z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.4z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.4z + z2 = 0 + 1
0.4z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.4z + z2 = 1

The z term is 0.4z.  Take half its coefficient (0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
0.4z + 0.04 + z2 = 1 + 0.04

Reorder the terms:
0.04 + 0.4z + z2 = 1 + 0.04

Combine like terms: 1 + 0.04 = 1.04
0.04 + 0.4z + z2 = 1.04

Factor a perfect square on the left side:
(z + 0.2)(z + 0.2) = 1.04

Calculate the square root of the right side: 1.019803903

Break this problem into two subproblems by setting 
(z + 0.2) equal to 1.019803903 and -1.019803903.

Subproblem 1
z + 0.2 = 1.019803903

Simplifying
z + 0.2 = 1.019803903

Reorder the terms:
0.2 + z = 1.019803903

Solving
0.2 + z = 1.019803903

Solving for variable 'z'.

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

Add '-0.2' to each side of the equation.
0.2 + -0.2 + z = 1.019803903 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + z = 1.019803903 + -0.2
z = 1.019803903 + -0.2

Combine like terms: 1.019803903 + -0.2 = 0.819803903
z = 0.819803903

Simplifying
z = 0.819803903

Subproblem 2
z + 0.2 = -1.019803903

Simplifying
z + 0.2 = -1.019803903

Reorder the terms:
0.2 + z = -1.019803903

Solving
0.2 + z = -1.019803903

Solving for variable 'z'.

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

Add '-0.2' to each side of the equation.
0.2 + -0.2 + z = -1.019803903 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + z = -1.019803903 + -0.2
z = -1.019803903 + -0.2

Combine like terms: -1.019803903 + -0.2 = -1.219803903
z = -1.219803903

Simplifying
z = -1.219803903

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.819803903, -1.219803903}