10(t)=-2t^2+20t+10

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Solution for 10(t)=-2t^2+20t+10 equation: 


Simplifying
10(t) = -2t2 + 20t + 10

Reorder the terms:
10t = 10 + 20t + -2t2

Solving
10t = 10 + 20t + -2t2

Solving for variable 't'.

Reorder the terms:
-10 + 10t + -20t + 2t2 = 10 + 20t + -2t2 + -10 + -20t + 2t2

Combine like terms: 10t + -20t = -10t
-10 + -10t + 2t2 = 10 + 20t + -2t2 + -10 + -20t + 2t2

Reorder the terms:
-10 + -10t + 2t2 = 10 + -10 + 20t + -20t + -2t2 + 2t2

Combine like terms: 10 + -10 = 0
-10 + -10t + 2t2 = 0 + 20t + -20t + -2t2 + 2t2
-10 + -10t + 2t2 = 20t + -20t + -2t2 + 2t2

Combine like terms: 20t + -20t = 0
-10 + -10t + 2t2 = 0 + -2t2 + 2t2
-10 + -10t + 2t2 = -2t2 + 2t2

Combine like terms: -2t2 + 2t2 = 0
-10 + -10t + 2t2 = 0

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

Ignore the factor 2.

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

Simplifying
-5 + -5t + t2 = 0

Solving
-5 + -5t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -5t + 5 + t2 = 0 + 5

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

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

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

The t term is -5t.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5t + 6.25 + t2 = 5 + 6.25

Reorder the terms:
6.25 + -5t + t2 = 5 + 6.25

Combine like terms: 5 + 6.25 = 11.25
6.25 + -5t + t2 = 11.25

Factor a perfect square on the left side:
(t + -2.5)(t + -2.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(t + -2.5) equal to 3.354101966 and -3.354101966.


Subproblem 1
t + -2.5 = 3.354101966

Simplifying
t + -2.5 = 3.354101966

Reorder the terms:
-2.5 + t = 3.354101966

Solving
-2.5 + t = 3.354101966

Solving for variable 't'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + t = 3.354101966 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + t = 3.354101966 + 2.5
t = 3.354101966 + 2.5

Combine like terms: 3.354101966 + 2.5 = 5.854101966
t = 5.854101966

Simplifying
t = 5.854101966


Subproblem 2
t + -2.5 = -3.354101966

Simplifying
t + -2.5 = -3.354101966

Reorder the terms:
-2.5 + t = -3.354101966

Solving
-2.5 + t = -3.354101966

Solving for variable 't'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + t = -3.354101966 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + t = -3.354101966 + 2.5
t = -3.354101966 + 2.5

Combine like terms: -3.354101966 + 2.5 = -0.854101966
t = -0.854101966

Simplifying
t = -0.854101966


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {5.854101966, -0.854101966}
Solution
t = {5.854101966, -0.854101966}

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