(t)=20t+2t^2-100

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Solution for (t)=20t+2t^2-100 equation:


Simplifying
(t) = 20t + 2t2 + -100
t = 20t + 2t2 + -100

Reorder the terms:
t = -100 + 20t + 2t2

Solving
t = -100 + 20t + 2t2

Solving for variable 't'.

Reorder the terms:
100 + t + -20t + -2t2 = -100 + 20t + 2t2 + 100 + -20t + -2t2

Combine like terms: t + -20t = -19t
100 + -19t + -2t2 = -100 + 20t + 2t2 + 100 + -20t + -2t2

Reorder the terms:
100 + -19t + -2t2 = -100 + 100 + 20t + -20t + 2t2 + -2t2

Combine like terms: -100 + 100 = 0
100 + -19t + -2t2 = 0 + 20t + -20t + 2t2 + -2t2
100 + -19t + -2t2 = 20t + -20t + 2t2 + -2t2

Combine like terms: 20t + -20t = 0
100 + -19t + -2t2 = 0 + 2t2 + -2t2
100 + -19t + -2t2 = 2t2 + -2t2

Combine like terms: 2t2 + -2t2 = 0
100 + -19t + -2t2 = 0

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

Divide each side by '-2'.
-50 + 9.5t + t2 = 0

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + 9.5t + 50 + t2 = 0 + 50

Reorder the terms:
-50 + 50 + 9.5t + t2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + 9.5t + t2 = 0 + 50
9.5t + t2 = 0 + 50

Combine like terms: 0 + 50 = 50
9.5t + t2 = 50

The t term is 9.5t.  Take half its coefficient (4.75).
Square it (22.5625) and add it to both sides.

Add '22.5625' to each side of the equation.
9.5t + 22.5625 + t2 = 50 + 22.5625

Reorder the terms:
22.5625 + 9.5t + t2 = 50 + 22.5625

Combine like terms: 50 + 22.5625 = 72.5625
22.5625 + 9.5t + t2 = 72.5625

Factor a perfect square on the left side:
(t + 4.75)(t + 4.75) = 72.5625

Calculate the square root of the right side: 8.518362519

Break this problem into two subproblems by setting 
(t + 4.75) equal to 8.518362519 and -8.518362519.

Subproblem 1

t + 4.75 = 8.518362519 Simplifying t + 4.75 = 8.518362519 Reorder the terms: 4.75 + t = 8.518362519 Solving 4.75 + t = 8.518362519 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + t = 8.518362519 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + t = 8.518362519 + -4.75 t = 8.518362519 + -4.75 Combine like terms: 8.518362519 + -4.75 = 3.768362519 t = 3.768362519 Simplifying t = 3.768362519

Subproblem 2

t + 4.75 = -8.518362519 Simplifying t + 4.75 = -8.518362519 Reorder the terms: 4.75 + t = -8.518362519 Solving 4.75 + t = -8.518362519 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + t = -8.518362519 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + t = -8.518362519 + -4.75 t = -8.518362519 + -4.75 Combine like terms: -8.518362519 + -4.75 = -13.268362519 t = -13.268362519 Simplifying t = -13.268362519

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.768362519, -13.268362519}

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