-5t^2-10t+250=0

Solution for -5t^2-10t+250=0 equation:

Simplifying
-5t2 + -10t + 250 = 0

Reorder the terms:
250 + -10t + -5t2 = 0

Solving
250 + -10t + -5t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '5'.
5(50 + -2t + -1t2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(50 + -2t + -1t2)' equal to zero and attempt to solve:

Simplifying
50 + -2t + -1t2 = 0

Solving
50 + -2t + -1t2 = 0

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

Divide each side by '-1'.
-50 + 2t + t2 = 0

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + 2t + 50 + t2 = 0 + 50

Reorder the terms:
-50 + 50 + 2t + t2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + 2t + t2 = 0 + 50
2t + t2 = 0 + 50

Combine like terms: 0 + 50 = 50
2t + t2 = 50

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 = 50 + 1

Reorder the terms:
1 + 2t + t2 = 50 + 1

Combine like terms: 50 + 1 = 51
1 + 2t + t2 = 51

Factor a perfect square on the left side:
(t + 1)(t + 1) = 51

Calculate the square root of the right side: 7.141428429

Break this problem into two subproblems by setting 
(t + 1) equal to 7.141428429 and -7.141428429.

Subproblem 1
t + 1 = 7.141428429

Simplifying
t + 1 = 7.141428429

Reorder the terms:
1 + t = 7.141428429

Solving
1 + t = 7.141428429

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 = 7.141428429 + -1

Combine like terms: 1 + -1 = 0
0 + t = 7.141428429 + -1
t = 7.141428429 + -1

Combine like terms: 7.141428429 + -1 = 6.141428429
t = 6.141428429

Simplifying
t = 6.141428429

Subproblem 2
t + 1 = -7.141428429

Simplifying
t + 1 = -7.141428429

Reorder the terms:
1 + t = -7.141428429

Solving
1 + t = -7.141428429

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 = -7.141428429 + -1

Combine like terms: 1 + -1 = 0
0 + t = -7.141428429 + -1
t = -7.141428429 + -1

Combine like terms: -7.141428429 + -1 = -8.141428429
t = -8.141428429

Simplifying
t = -8.141428429

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {6.141428429, -8.141428429}
Solution
t = {6.141428429, -8.141428429}