20t-0.1t^2=400

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Solution for 20t-0.1t^2=400 equation:


Simplifying
20t + -0.1t2 = 400

Solving
20t + -0.1t2 = 400

Solving for variable 't'.

Reorder the terms:
-400 + 20t + -0.1t2 = 400 + -400

Combine like terms: 400 + -400 = 0
-400 + 20t + -0.1t2 = 0

Begin completing the square.  Divide all terms by
-0.1 the coefficient of the squared term: 

Divide each side by '-0.1'.
4000 + -200t + t2 = 0

Move the constant term to the right:

Add '-4000' to each side of the equation.
4000 + -200t + -4000 + t2 = 0 + -4000

Reorder the terms:
4000 + -4000 + -200t + t2 = 0 + -4000

Combine like terms: 4000 + -4000 = 0
0 + -200t + t2 = 0 + -4000
-200t + t2 = 0 + -4000

Combine like terms: 0 + -4000 = -4000
-200t + t2 = -4000

The t term is -200t.  Take half its coefficient (-100).
Square it (10000) and add it to both sides.

Add '10000' to each side of the equation.
-200t + 10000 + t2 = -4000 + 10000

Reorder the terms:
10000 + -200t + t2 = -4000 + 10000

Combine like terms: -4000 + 10000 = 6000
10000 + -200t + t2 = 6000

Factor a perfect square on the left side:
(t + -100)(t + -100) = 6000

Calculate the square root of the right side: 77.459666924

Break this problem into two subproblems by setting 
(t + -100) equal to 77.459666924 and -77.459666924.

Subproblem 1

t + -100 = 77.459666924 Simplifying t + -100 = 77.459666924 Reorder the terms: -100 + t = 77.459666924 Solving -100 + t = 77.459666924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '100' to each side of the equation. -100 + 100 + t = 77.459666924 + 100 Combine like terms: -100 + 100 = 0 0 + t = 77.459666924 + 100 t = 77.459666924 + 100 Combine like terms: 77.459666924 + 100 = 177.459666924 t = 177.459666924 Simplifying t = 177.459666924

Subproblem 2

t + -100 = -77.459666924 Simplifying t + -100 = -77.459666924 Reorder the terms: -100 + t = -77.459666924 Solving -100 + t = -77.459666924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '100' to each side of the equation. -100 + 100 + t = -77.459666924 + 100 Combine like terms: -100 + 100 = 0 0 + t = -77.459666924 + 100 t = -77.459666924 + 100 Combine like terms: -77.459666924 + 100 = 22.540333076 t = 22.540333076 Simplifying t = 22.540333076

Solution

The solution to the problem is based on the solutions from the subproblems. t = {177.459666924, 22.540333076}

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