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