S^2+4S-24=0

Solution for S^2+4S-24=0 equation:

Simplifying
S2 + 4S + -24 = 0

Reorder the terms:
-24 + 4S + S2 = 0

Solving
-24 + 4S + S2 = 0

Solving for variable 'S'.

Begin completing the square.

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + 4S + 24 + S2 = 0 + 24

Reorder the terms:
-24 + 24 + 4S + S2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + 4S + S2 = 0 + 24
4S + S2 = 0 + 24

Combine like terms: 0 + 24 = 24
4S + S2 = 24

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 = 24 + 4

Reorder the terms:
4 + 4S + S2 = 24 + 4

Combine like terms: 24 + 4 = 28
4 + 4S + S2 = 28

Factor a perfect square on the left side:
(S + 2)(S + 2) = 28

Calculate the square root of the right side: 5.291502622

Break this problem into two subproblems by setting 
(S + 2) equal to 5.291502622 and -5.291502622.

Subproblem 1
S + 2 = 5.291502622

Simplifying
S + 2 = 5.291502622

Reorder the terms:
2 + S = 5.291502622

Solving
2 + S = 5.291502622

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 = 5.291502622 + -2

Combine like terms: 2 + -2 = 0
0 + S = 5.291502622 + -2
S = 5.291502622 + -2

Combine like terms: 5.291502622 + -2 = 3.291502622
S = 3.291502622

Simplifying
S = 3.291502622

Subproblem 2
S + 2 = -5.291502622

Simplifying
S + 2 = -5.291502622

Reorder the terms:
2 + S = -5.291502622

Solving
2 + S = -5.291502622

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 = -5.291502622 + -2

Combine like terms: 2 + -2 = 0
0 + S = -5.291502622 + -2
S = -5.291502622 + -2

Combine like terms: -5.291502622 + -2 = -7.291502622
S = -7.291502622

Simplifying
S = -7.291502622

Solution
The solution to the problem is based on the solutions
from the subproblems.
S = {3.291502622, -7.291502622}