-200t^2+3600t-6400=8500

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Solution for -200t^2+3600t-6400=8500 equation: 


Simplifying
-200t2 + 3600t + -6400 = 8500

Reorder the terms:
-6400 + 3600t + -200t2 = 8500

Solving
-6400 + 3600t + -200t2 = 8500

Solving for variable 't'.

Reorder the terms:
-6400 + -8500 + 3600t + -200t2 = 8500 + -8500

Combine like terms: -6400 + -8500 = -14900
-14900 + 3600t + -200t2 = 8500 + -8500

Combine like terms: 8500 + -8500 = 0
-14900 + 3600t + -200t2 = 0

Factor out the Greatest Common Factor (GCF), '100'.
100(-149 + 36t + -2t2) = 0

Ignore the factor 100.

Subproblem 1
Set the factor '(-149 + 36t + -2t2)' equal to zero and attempt to solve:

Simplifying
-149 + 36t + -2t2 = 0

Solving
-149 + 36t + -2t2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
74.5 + -18t + t2 = 0

Move the constant term to the right:

Add '-74.5' to each side of the equation.
74.5 + -18t + -74.5 + t2 = 0 + -74.5

Reorder the terms:
74.5 + -74.5 + -18t + t2 = 0 + -74.5

Combine like terms: 74.5 + -74.5 = 0.0
0.0 + -18t + t2 = 0 + -74.5
-18t + t2 = 0 + -74.5

Combine like terms: 0 + -74.5 = -74.5
-18t + t2 = -74.5

The t term is -18t.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18t + 81 + t2 = -74.5 + 81

Reorder the terms:
81 + -18t + t2 = -74.5 + 81

Combine like terms: -74.5 + 81 = 6.5
81 + -18t + t2 = 6.5

Factor a perfect square on the left side:
(t + -9)(t + -9) = 6.5

Calculate the square root of the right side: 2.549509757

Break this problem into two subproblems by setting 
(t + -9) equal to 2.549509757 and -2.549509757.


Subproblem 1
t + -9 = 2.549509757

Simplifying
t + -9 = 2.549509757

Reorder the terms:
-9 + t = 2.549509757

Solving
-9 + t = 2.549509757

Solving for variable 't'.

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

Add '9' to each side of the equation.
-9 + 9 + t = 2.549509757 + 9

Combine like terms: -9 + 9 = 0
0 + t = 2.549509757 + 9
t = 2.549509757 + 9

Combine like terms: 2.549509757 + 9 = 11.549509757
t = 11.549509757

Simplifying
t = 11.549509757


Subproblem 2
t + -9 = -2.549509757

Simplifying
t + -9 = -2.549509757

Reorder the terms:
-9 + t = -2.549509757

Solving
-9 + t = -2.549509757

Solving for variable 't'.

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

Add '9' to each side of the equation.
-9 + 9 + t = -2.549509757 + 9

Combine like terms: -9 + 9 = 0
0 + t = -2.549509757 + 9
t = -2.549509757 + 9

Combine like terms: -2.549509757 + 9 = 6.450490243
t = 6.450490243

Simplifying
t = 6.450490243


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {11.549509757, 6.450490243}
Solution
t = {11.549509757, 6.450490243}

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