z^2+3z=29

Solution for z^2+3z=29 equation:

Simplifying
z2 + 3z = 29

Reorder the terms:
3z + z2 = 29

Solving
3z + z2 = 29

Solving for variable 'z'.

Reorder the terms:
-29 + 3z + z2 = 29 + -29

Combine like terms: 29 + -29 = 0
-29 + 3z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '29' to each side of the equation.
-29 + 3z + 29 + z2 = 0 + 29

Reorder the terms:
-29 + 29 + 3z + z2 = 0 + 29

Combine like terms: -29 + 29 = 0
0 + 3z + z2 = 0 + 29
3z + z2 = 0 + 29

Combine like terms: 0 + 29 = 29
3z + z2 = 29

The z term is 3z.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3z + 2.25 + z2 = 29 + 2.25

Reorder the terms:
2.25 + 3z + z2 = 29 + 2.25

Combine like terms: 29 + 2.25 = 31.25
2.25 + 3z + z2 = 31.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 31.25

Calculate the square root of the right side: 5.590169944

Break this problem into two subproblems by setting 
(z + 1.5) equal to 5.590169944 and -5.590169944.

Subproblem 1
z + 1.5 = 5.590169944

Simplifying
z + 1.5 = 5.590169944

Reorder the terms:
1.5 + z = 5.590169944

Solving
1.5 + z = 5.590169944

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z = 5.590169944 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = 5.590169944 + -1.5
z = 5.590169944 + -1.5

Combine like terms: 5.590169944 + -1.5 = 4.090169944
z = 4.090169944

Simplifying
z = 4.090169944

Subproblem 2
z + 1.5 = -5.590169944

Simplifying
z + 1.5 = -5.590169944

Reorder the terms:
1.5 + z = -5.590169944

Solving
1.5 + z = -5.590169944

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z = -5.590169944 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = -5.590169944 + -1.5
z = -5.590169944 + -1.5

Combine like terms: -5.590169944 + -1.5 = -7.090169944
z = -7.090169944

Simplifying
z = -7.090169944

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.090169944, -7.090169944}