z^2+2z-13=0

Solution for z^2+2z-13=0 equation:

Simplifying
z2 + 2z + -13 = 0

Reorder the terms:
-13 + 2z + z2 = 0

Solving
-13 + 2z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + 2z + 13 + z2 = 0 + 13

Reorder the terms:
-13 + 13 + 2z + z2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + 2z + z2 = 0 + 13
2z + z2 = 0 + 13

Combine like terms: 0 + 13 = 13
2z + z2 = 13

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 = 13 + 1

Reorder the terms:
1 + 2z + z2 = 13 + 1

Combine like terms: 13 + 1 = 14
1 + 2z + z2 = 14

Factor a perfect square on the left side:
(z + 1)(z + 1) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(z + 1) equal to 3.741657387 and -3.741657387.

Subproblem 1
z + 1 = 3.741657387

Simplifying
z + 1 = 3.741657387

Reorder the terms:
1 + z = 3.741657387

Solving
1 + z = 3.741657387

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 = 3.741657387 + -1

Combine like terms: 1 + -1 = 0
0 + z = 3.741657387 + -1
z = 3.741657387 + -1

Combine like terms: 3.741657387 + -1 = 2.741657387
z = 2.741657387

Simplifying
z = 2.741657387

Subproblem 2
z + 1 = -3.741657387

Simplifying
z + 1 = -3.741657387

Reorder the terms:
1 + z = -3.741657387

Solving
1 + z = -3.741657387

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 = -3.741657387 + -1

Combine like terms: 1 + -1 = 0
0 + z = -3.741657387 + -1
z = -3.741657387 + -1

Combine like terms: -3.741657387 + -1 = -4.741657387
z = -4.741657387

Simplifying
z = -4.741657387

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.741657387, -4.741657387}