50=5t+20t^2

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Solution for 50=5t+20t^2 equation:


Simplifying
50 = 5t + 20t2

Solving
50 = 5t + 20t2

Solving for variable 't'.

Reorder the terms:
50 + -5t + -20t2 = 5t + -5t + 20t2 + -20t2

Combine like terms: 5t + -5t = 0
50 + -5t + -20t2 = 0 + 20t2 + -20t2
50 + -5t + -20t2 = 20t2 + -20t2

Combine like terms: 20t2 + -20t2 = 0
50 + -5t + -20t2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(10 + -1t + -4t2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(10 + -1t + -4t2)' equal to zero and attempt to solve:

Simplifying
10 + -1t + -4t2 = 0

Solving
10 + -1t + -4t2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-2.5 + 0.25t + t2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 0.25t + 2.5 + t2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 0.25t + t2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 0.25t + t2 = 0 + 2.5
0.25t + t2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
0.25t + t2 = 2.5

The t term is 0.25t.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25t + 0.015625 + t2 = 2.5 + 0.015625

Reorder the terms:
0.015625 + 0.25t + t2 = 2.5 + 0.015625

Combine like terms: 2.5 + 0.015625 = 2.515625
0.015625 + 0.25t + t2 = 2.515625

Factor a perfect square on the left side:
(t + 0.125)(t + 0.125) = 2.515625

Calculate the square root of the right side: 1.586072193

Break this problem into two subproblems by setting 
(t + 0.125) equal to 1.586072193 and -1.586072193.


Subproblem 1
t + 0.125 = 1.586072193

Simplifying
t + 0.125 = 1.586072193

Reorder the terms:
0.125 + t = 1.586072193

Solving
0.125 + t = 1.586072193

Solving for variable 't'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + t = 1.586072193 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + t = 1.586072193 + -0.125
t = 1.586072193 + -0.125

Combine like terms: 1.586072193 + -0.125 = 1.461072193
t = 1.461072193

Simplifying
t = 1.461072193


Subproblem 2
t + 0.125 = -1.586072193

Simplifying
t + 0.125 = -1.586072193

Reorder the terms:
0.125 + t = -1.586072193

Solving
0.125 + t = -1.586072193

Solving for variable 't'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + t = -1.586072193 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + t = -1.586072193 + -0.125
t = -1.586072193 + -0.125

Combine like terms: -1.586072193 + -0.125 = -1.711072193
t = -1.711072193

Simplifying
t = -1.711072193


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.461072193, -1.711072193}
Solution
t = {1.461072193, -1.711072193}

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