z^2+2z=10

Solution for z^2+2z=10 equation:

Simplifying
z2 + 2z = 10

Reorder the terms:
2z + z2 = 10

Solving
2z + z2 = 10

Solving for variable 'z'.

Reorder the terms:
-10 + 2z + z2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 2z + 10 + z2 = 0 + 10

Reorder the terms:
-10 + 10 + 2z + z2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 2z + z2 = 0 + 10
2z + z2 = 0 + 10

Combine like terms: 0 + 10 = 10
2z + z2 = 10

The z term is 2z.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2z + 1 + z2 = 10 + 1

Reorder the terms:
1 + 2z + z2 = 10 + 1

Combine like terms: 10 + 1 = 11
1 + 2z + z2 = 11

Factor a perfect square on the left side:
(z + 1)(z + 1) = 11

Calculate the square root of the right side: 3.31662479

Break this problem into two subproblems by setting 
(z + 1) equal to 3.31662479 and -3.31662479.

Subproblem 1
z + 1 = 3.31662479

Simplifying
z + 1 = 3.31662479

Reorder the terms:
1 + z = 3.31662479

Solving
1 + z = 3.31662479

Solving for variable 'z'.

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

Add '-1' to each side of the equation.
1 + -1 + z = 3.31662479 + -1

Combine like terms: 1 + -1 = 0
0 + z = 3.31662479 + -1
z = 3.31662479 + -1

Combine like terms: 3.31662479 + -1 = 2.31662479
z = 2.31662479

Simplifying
z = 2.31662479

Subproblem 2
z + 1 = -3.31662479

Simplifying
z + 1 = -3.31662479

Reorder the terms:
1 + z = -3.31662479

Solving
1 + z = -3.31662479

Solving for variable 'z'.

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

Add '-1' to each side of the equation.
1 + -1 + z = -3.31662479 + -1

Combine like terms: 1 + -1 = 0
0 + z = -3.31662479 + -1
z = -3.31662479 + -1

Combine like terms: -3.31662479 + -1 = -4.31662479
z = -4.31662479

Simplifying
z = -4.31662479

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.31662479, -4.31662479}