2z^2+8z-3=0

Solution for 2z^2+8z-3=0 equation:

Simplifying
2z2 + 8z + -3 = 0

Reorder the terms:
-3 + 8z + 2z2 = 0

Solving
-3 + 8z + 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'.
-1.5 + 4z + z2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 4z + 1.5 + z2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 4z + z2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 4z + z2 = 0 + 1.5
4z + z2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
4z + z2 = 1.5

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 = 1.5 + 4

Reorder the terms:
4 + 4z + z2 = 1.5 + 4

Combine like terms: 1.5 + 4 = 5.5
4 + 4z + z2 = 5.5

Factor a perfect square on the left side:
(z + 2)(z + 2) = 5.5

Calculate the square root of the right side: 2.34520788

Break this problem into two subproblems by setting 
(z + 2) equal to 2.34520788 and -2.34520788.

Subproblem 1
z + 2 = 2.34520788

Simplifying
z + 2 = 2.34520788

Reorder the terms:
2 + z = 2.34520788

Solving
2 + z = 2.34520788

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 = 2.34520788 + -2

Combine like terms: 2 + -2 = 0
0 + z = 2.34520788 + -2
z = 2.34520788 + -2

Combine like terms: 2.34520788 + -2 = 0.34520788
z = 0.34520788

Simplifying
z = 0.34520788

Subproblem 2
z + 2 = -2.34520788

Simplifying
z + 2 = -2.34520788

Reorder the terms:
2 + z = -2.34520788

Solving
2 + z = -2.34520788

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 = -2.34520788 + -2

Combine like terms: 2 + -2 = 0
0 + z = -2.34520788 + -2
z = -2.34520788 + -2

Combine like terms: -2.34520788 + -2 = -4.34520788
z = -4.34520788

Simplifying
z = -4.34520788

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.34520788, -4.34520788}