2z^2+13z+2=0

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

Simplifying
2z2 + 13z + 2 = 0

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

Solving
2 + 13z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
1 + 6.5z + z2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 6.5z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 6.5z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 6.5z + z2 = 0 + -1
6.5z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
6.5z + z2 = -1

The z term is 6.5z.  Take half its coefficient (3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
6.5z + 10.5625 + z2 = -1 + 10.5625

Reorder the terms:
10.5625 + 6.5z + z2 = -1 + 10.5625

Combine like terms: -1 + 10.5625 = 9.5625
10.5625 + 6.5z + z2 = 9.5625

Factor a perfect square on the left side:
(z + 3.25)(z + 3.25) = 9.5625

Calculate the square root of the right side: 3.092329219

Break this problem into two subproblems by setting 
(z + 3.25) equal to 3.092329219 and -3.092329219.

Subproblem 1
z + 3.25 = 3.092329219

Simplifying
z + 3.25 = 3.092329219

Reorder the terms:
3.25 + z = 3.092329219

Solving
3.25 + z = 3.092329219

Solving for variable 'z'.

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

Add '-3.25' to each side of the equation.
3.25 + -3.25 + z = 3.092329219 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + z = 3.092329219 + -3.25
z = 3.092329219 + -3.25

Combine like terms: 3.092329219 + -3.25 = -0.157670781
z = -0.157670781

Simplifying
z = -0.157670781

Subproblem 2
z + 3.25 = -3.092329219

Simplifying
z + 3.25 = -3.092329219

Reorder the terms:
3.25 + z = -3.092329219

Solving
3.25 + z = -3.092329219

Solving for variable 'z'.

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

Add '-3.25' to each side of the equation.
3.25 + -3.25 + z = -3.092329219 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + z = -3.092329219 + -3.25
z = -3.092329219 + -3.25

Combine like terms: -3.092329219 + -3.25 = -6.342329219
z = -6.342329219

Simplifying
z = -6.342329219

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.157670781, -6.342329219}