.5(12-4s^2)-2s(1-5)=8

Solution for .5(12-4s^2)-2s(1-5)=8 equation:

Simplifying
0.5(12 + -4s2) + -2s(1 + -5) = 8
(12 * 0.5 + -4s2 * 0.5) + -2s(1 + -5) = 8
(6 + -2s2) + -2s(1 + -5) = 8

Combine like terms: 1 + -5 = -4
6 + -2s2 + -2s(-4) = 8

Reorder the terms for easier multiplication:
6 + -2s2 + -2 * -4s = 8

Multiply -2 * -4
6 + -2s2 + 8s = 8

Reorder the terms:
6 + 8s + -2s2 = 8

Solving
6 + 8s + -2s2 = 8

Solving for variable 's'.

Reorder the terms:
6 + -8 + 8s + -2s2 = 8 + -8

Combine like terms: 6 + -8 = -2
-2 + 8s + -2s2 = 8 + -8

Combine like terms: 8 + -8 = 0
-2 + 8s + -2s2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + 4s + -1s2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-1 + 4s + -1s2)' equal to zero and attempt to solve:

Simplifying
-1 + 4s + -1s2 = 0

Solving
-1 + 4s + -1s2 = 0

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

Divide each side by '-1'.
1 + -4s + s2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4s + -1 + s2 = 0 + -1

Reorder the terms:
1 + -1 + -4s + s2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4s + s2 = 0 + -1
-4s + s2 = 0 + -1

Combine like terms: 0 + -1 = -1
-4s + s2 = -1

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

Add '4' to each side of the equation.
-4s + 4 + s2 = -1 + 4

Reorder the terms:
4 + -4s + s2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + -4s + s2 = 3

Factor a perfect square on the left side:
(s + -2)(s + -2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(s + -2) equal to 1.732050808 and -1.732050808.

Subproblem 1
s + -2 = 1.732050808

Simplifying
s + -2 = 1.732050808

Reorder the terms:
-2 + s = 1.732050808

Solving
-2 + s = 1.732050808

Solving for variable 's'.

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

Add '2' to each side of the equation.
-2 + 2 + s = 1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + s = 1.732050808 + 2
s = 1.732050808 + 2

Combine like terms: 1.732050808 + 2 = 3.732050808
s = 3.732050808

Simplifying
s = 3.732050808

Subproblem 2
s + -2 = -1.732050808

Simplifying
s + -2 = -1.732050808

Reorder the terms:
-2 + s = -1.732050808

Solving
-2 + s = -1.732050808

Solving for variable 's'.

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

Add '2' to each side of the equation.
-2 + 2 + s = -1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + s = -1.732050808 + 2
s = -1.732050808 + 2

Combine like terms: -1.732050808 + 2 = 0.267949192
s = 0.267949192

Simplifying
s = 0.267949192

Solution
The solution to the problem is based on the solutions
from the subproblems.
s = {3.732050808, 0.267949192}
Solution
s = {3.732050808, 0.267949192}