z^2+3z=3

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

Simplifying
z2 + 3z = 3

Reorder the terms:
3z + z2 = 3

Solving
3z + z2 = 3

Solving for variable 'z'.

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 3 + 2.25

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

Combine like terms: 3 + 2.25 = 5.25
2.25 + 3z + z2 = 5.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(z + 1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
z + 1.5 = 2.291287847

Simplifying
z + 1.5 = 2.291287847

Reorder the terms:
1.5 + z = 2.291287847

Solving
1.5 + z = 2.291287847

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 = 2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = 2.291287847 + -1.5
z = 2.291287847 + -1.5

Combine like terms: 2.291287847 + -1.5 = 0.791287847
z = 0.791287847

Simplifying
z = 0.791287847

Subproblem 2
z + 1.5 = -2.291287847

Simplifying
z + 1.5 = -2.291287847

Reorder the terms:
1.5 + z = -2.291287847

Solving
1.5 + z = -2.291287847

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 = -2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = -2.291287847 + -1.5
z = -2.291287847 + -1.5

Combine like terms: -2.291287847 + -1.5 = -3.791287847
z = -3.791287847

Simplifying
z = -3.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.791287847, -3.791287847}