z+25=z^2-25

Solution for z+25=z^2-25 equation:

Simplifying
z + 25 = z2 + -25

Reorder the terms:
25 + z = z2 + -25

Reorder the terms:
25 + z = -25 + z2

Solving
25 + z = -25 + z2

Solving for variable 'z'.

Reorder the terms:
25 + 25 + z + -1z2 = -25 + z2 + 25 + -1z2

Combine like terms: 25 + 25 = 50
50 + z + -1z2 = -25 + z2 + 25 + -1z2

Reorder the terms:
50 + z + -1z2 = -25 + 25 + z2 + -1z2

Combine like terms: -25 + 25 = 0
50 + z + -1z2 = 0 + z2 + -1z2
50 + z + -1z2 = z2 + -1z2

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

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

Divide each side by '-1'.
-50 + -1z + z2 = 0

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + -1z + 50 + z2 = 0 + 50

Reorder the terms:
-50 + 50 + -1z + z2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + -1z + z2 = 0 + 50
-1z + z2 = 0 + 50

Combine like terms: 0 + 50 = 50
-1z + z2 = 50

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 50 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 50 + 0.25

Combine like terms: 50 + 0.25 = 50.25
0.25 + -1z + z2 = 50.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 50.25

Calculate the square root of the right side: 7.088723439

Break this problem into two subproblems by setting 
(z + 0.5) equal to 7.088723439 and -7.088723439.

Subproblem 1
z + 0.5 = 7.088723439

Simplifying
z + 0.5 = 7.088723439

Reorder the terms:
0.5 + z = 7.088723439

Solving
0.5 + z = 7.088723439

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = 7.088723439 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 7.088723439 + -0.5
z = 7.088723439 + -0.5

Combine like terms: 7.088723439 + -0.5 = 6.588723439
z = 6.588723439

Simplifying
z = 6.588723439

Subproblem 2
z + 0.5 = -7.088723439

Simplifying
z + 0.5 = -7.088723439

Reorder the terms:
0.5 + z = -7.088723439

Solving
0.5 + z = -7.088723439

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = -7.088723439 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -7.088723439 + -0.5
z = -7.088723439 + -0.5

Combine like terms: -7.088723439 + -0.5 = -7.588723439
z = -7.588723439

Simplifying
z = -7.588723439

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {6.588723439, -7.588723439}