78=0.05s^2+1.1s

Solution for 78=0.05s^2+1.1s equation:

Simplifying
78 = 0.05s2 + 1.1s

Reorder the terms:
78 = 1.1s + 0.05s2

Solving
78 = 1.1s + 0.05s2

Solving for variable 's'.

Reorder the terms:
78 + -1.1s + -0.05s2 = 1.1s + -1.1s + 0.05s2 + -0.05s2

Combine like terms: 1.1s + -1.1s = 0.0
78 + -1.1s + -0.05s2 = 0.0 + 0.05s2 + -0.05s2
78 + -1.1s + -0.05s2 = 0.05s2 + -0.05s2

Combine like terms: 0.05s2 + -0.05s2 = 0.00
78 + -1.1s + -0.05s2 = 0.00

Begin completing the square.  Divide all terms by
-0.05 the coefficient of the squared term: 

Divide each side by '-0.05'.
-1560 + 22s + s2 = 0

Move the constant term to the right:

Add '1560' to each side of the equation.
-1560 + 22s + 1560 + s2 = 0 + 1560

Reorder the terms:
-1560 + 1560 + 22s + s2 = 0 + 1560

Combine like terms: -1560 + 1560 = 0
0 + 22s + s2 = 0 + 1560
22s + s2 = 0 + 1560

Combine like terms: 0 + 1560 = 1560
22s + s2 = 1560

The s term is 22s.  Take half its coefficient (11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
22s + 121 + s2 = 1560 + 121

Reorder the terms:
121 + 22s + s2 = 1560 + 121

Combine like terms: 1560 + 121 = 1681
121 + 22s + s2 = 1681

Factor a perfect square on the left side:
(s + 11)(s + 11) = 1681

Calculate the square root of the right side: 41

Break this problem into two subproblems by setting 
(s + 11) equal to 41 and -41.

Subproblem 1
s + 11 = 41

Simplifying
s + 11 = 41

Reorder the terms:
11 + s = 41

Solving
11 + s = 41

Solving for variable 's'.

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

Add '-11' to each side of the equation.
11 + -11 + s = 41 + -11

Combine like terms: 11 + -11 = 0
0 + s = 41 + -11
s = 41 + -11

Combine like terms: 41 + -11 = 30
s = 30

Simplifying
s = 30

Subproblem 2
s + 11 = -41

Simplifying
s + 11 = -41

Reorder the terms:
11 + s = -41

Solving
11 + s = -41

Solving for variable 's'.

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

Add '-11' to each side of the equation.
11 + -11 + s = -41 + -11

Combine like terms: 11 + -11 = 0
0 + s = -41 + -11
s = -41 + -11

Combine like terms: -41 + -11 = -52
s = -52

Simplifying
s = -52

Solution
The solution to the problem is based on the solutions
from the subproblems.
s = {30, -52}