0=65-10t-5t^2

Solution for 0=65-10t-5t^2 equation:

Simplifying
0 = 65 + -10t + -5t2

Solving
0 = 65 + -10t + -5t2

Solving for variable 't'.

Combine like terms: 0 + -65 = -65
-65 + 10t + 5t2 = 65 + -10t + -5t2 + -65 + 10t + 5t2

Reorder the terms:
-65 + 10t + 5t2 = 65 + -65 + -10t + 10t + -5t2 + 5t2

Combine like terms: 65 + -65 = 0
-65 + 10t + 5t2 = 0 + -10t + 10t + -5t2 + 5t2
-65 + 10t + 5t2 = -10t + 10t + -5t2 + 5t2

Combine like terms: -10t + 10t = 0
-65 + 10t + 5t2 = 0 + -5t2 + 5t2
-65 + 10t + 5t2 = -5t2 + 5t2

Combine like terms: -5t2 + 5t2 = 0
-65 + 10t + 5t2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-13 + 2t + t2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-13 + 2t + t2)' equal to zero and attempt to solve:

Simplifying
-13 + 2t + t2 = 0

Solving
-13 + 2t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + 2t + 13 + t2 = 0 + 13

Reorder the terms:
-13 + 13 + 2t + t2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + 2t + t2 = 0 + 13
2t + t2 = 0 + 13

Combine like terms: 0 + 13 = 13
2t + t2 = 13

The t term is 2t.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2t + 1 + t2 = 13 + 1

Reorder the terms:
1 + 2t + t2 = 13 + 1

Combine like terms: 13 + 1 = 14
1 + 2t + t2 = 14

Factor a perfect square on the left side:
(t + 1)(t + 1) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(t + 1) equal to 3.741657387 and -3.741657387.

Subproblem 1
t + 1 = 3.741657387

Simplifying
t + 1 = 3.741657387

Reorder the terms:
1 + t = 3.741657387

Solving
1 + t = 3.741657387

Solving for variable 't'.

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

Add '-1' to each side of the equation.
1 + -1 + t = 3.741657387 + -1

Combine like terms: 1 + -1 = 0
0 + t = 3.741657387 + -1
t = 3.741657387 + -1

Combine like terms: 3.741657387 + -1 = 2.741657387
t = 2.741657387

Simplifying
t = 2.741657387

Subproblem 2
t + 1 = -3.741657387

Simplifying
t + 1 = -3.741657387

Reorder the terms:
1 + t = -3.741657387

Solving
1 + t = -3.741657387

Solving for variable 't'.

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

Add '-1' to each side of the equation.
1 + -1 + t = -3.741657387 + -1

Combine like terms: 1 + -1 = 0
0 + t = -3.741657387 + -1
t = -3.741657387 + -1

Combine like terms: -3.741657387 + -1 = -4.741657387
t = -4.741657387

Simplifying
t = -4.741657387

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.741657387, -4.741657387}
Solution
t = {2.741657387, -4.741657387}