0=z^2+4z-3

Solution for 0=z^2+4z-3 equation:

Simplifying
0 = z2 + 4z + -3

Reorder the terms:
0 = -3 + 4z + z2

Solving
0 = -3 + 4z + z2

Solving for variable 'z'.

Combine like terms: 0 + 3 = 3
3 + -4z + -1z2 = -3 + 4z + z2 + 3 + -4z + -1z2

Reorder the terms:
3 + -4z + -1z2 = -3 + 3 + 4z + -4z + z2 + -1z2

Combine like terms: -3 + 3 = 0
3 + -4z + -1z2 = 0 + 4z + -4z + z2 + -1z2
3 + -4z + -1z2 = 4z + -4z + z2 + -1z2

Combine like terms: 4z + -4z = 0
3 + -4z + -1z2 = 0 + z2 + -1z2
3 + -4z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
3 + -4z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-3 + 4z + z2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 4z + 3 + z2 = 0 + 3

Reorder the terms:
-3 + 3 + 4z + z2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 4z + z2 = 0 + 3
4z + z2 = 0 + 3

Combine like terms: 0 + 3 = 3
4z + z2 = 3

The z term is 4z.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z + 4 + z2 = 3 + 4

Reorder the terms:
4 + 4z + z2 = 3 + 4

Combine like terms: 3 + 4 = 7
4 + 4z + z2 = 7

Factor a perfect square on the left side:
(z + 2)(z + 2) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(z + 2) equal to 2.645751311 and -2.645751311.

Subproblem 1
z + 2 = 2.645751311

Simplifying
z + 2 = 2.645751311

Reorder the terms:
2 + z = 2.645751311

Solving
2 + z = 2.645751311

Solving for variable 'z'.

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

Add '-2' to each side of the equation.
2 + -2 + z = 2.645751311 + -2

Combine like terms: 2 + -2 = 0
0 + z = 2.645751311 + -2
z = 2.645751311 + -2

Combine like terms: 2.645751311 + -2 = 0.645751311
z = 0.645751311

Simplifying
z = 0.645751311

Subproblem 2
z + 2 = -2.645751311

Simplifying
z + 2 = -2.645751311

Reorder the terms:
2 + z = -2.645751311

Solving
2 + z = -2.645751311

Solving for variable 'z'.

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

Add '-2' to each side of the equation.
2 + -2 + z = -2.645751311 + -2

Combine like terms: 2 + -2 = 0
0 + z = -2.645751311 + -2
z = -2.645751311 + -2

Combine like terms: -2.645751311 + -2 = -4.645751311
z = -4.645751311

Simplifying
z = -4.645751311

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.645751311, -4.645751311}