4z+8z^2=90

Solution for 4z+8z^2=90 equation:

Simplifying
4z + 8z2 = 90

Solving
4z + 8z2 = 90

Solving for variable 'z'.

Reorder the terms:
-90 + 4z + 8z2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 4z + 8z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-45 + 2z + 4z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-45 + 2z + 4z2)' equal to zero and attempt to solve:

Simplifying
-45 + 2z + 4z2 = 0

Solving
-45 + 2z + 4z2 = 0

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

Divide each side by '4'.
-11.25 + 0.5z + z2 = 0

Move the constant term to the right:

Add '11.25' to each side of the equation.
-11.25 + 0.5z + 11.25 + z2 = 0 + 11.25

Reorder the terms:
-11.25 + 11.25 + 0.5z + z2 = 0 + 11.25

Combine like terms: -11.25 + 11.25 = 0.00
0.00 + 0.5z + z2 = 0 + 11.25
0.5z + z2 = 0 + 11.25

Combine like terms: 0 + 11.25 = 11.25
0.5z + z2 = 11.25

The z term is 0.5z.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5z + 0.0625 + z2 = 11.25 + 0.0625

Reorder the terms:
0.0625 + 0.5z + z2 = 11.25 + 0.0625

Combine like terms: 11.25 + 0.0625 = 11.3125
0.0625 + 0.5z + z2 = 11.3125

Factor a perfect square on the left side:
(z + 0.25)(z + 0.25) = 11.3125

Calculate the square root of the right side: 3.363406012

Break this problem into two subproblems by setting 
(z + 0.25) equal to 3.363406012 and -3.363406012.

Subproblem 1
z + 0.25 = 3.363406012

Simplifying
z + 0.25 = 3.363406012

Reorder the terms:
0.25 + z = 3.363406012

Solving
0.25 + z = 3.363406012

Solving for variable 'z'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + z = 3.363406012 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + z = 3.363406012 + -0.25
z = 3.363406012 + -0.25

Combine like terms: 3.363406012 + -0.25 = 3.113406012
z = 3.113406012

Simplifying
z = 3.113406012

Subproblem 2
z + 0.25 = -3.363406012

Simplifying
z + 0.25 = -3.363406012

Reorder the terms:
0.25 + z = -3.363406012

Solving
0.25 + z = -3.363406012

Solving for variable 'z'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + z = -3.363406012 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + z = -3.363406012 + -0.25
z = -3.363406012 + -0.25

Combine like terms: -3.363406012 + -0.25 = -3.613406012
z = -3.613406012

Simplifying
z = -3.613406012

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.113406012, -3.613406012}
Solution
z = {3.113406012, -3.613406012}