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Simplifying 8500 = -200t2 + 3600t + -6400 Reorder the terms: 8500 = -6400 + 3600t + -200t2 Solving 8500 = -6400 + 3600t + -200t2 Solving for variable 't'. Combine like terms: 8500 + 6400 = 14900 14900 + -3600t + 200t2 = -6400 + 3600t + -200t2 + 6400 + -3600t + 200t2 Reorder the terms: 14900 + -3600t + 200t2 = -6400 + 6400 + 3600t + -3600t + -200t2 + 200t2 Combine like terms: -6400 + 6400 = 0 14900 + -3600t + 200t2 = 0 + 3600t + -3600t + -200t2 + 200t2 14900 + -3600t + 200t2 = 3600t + -3600t + -200t2 + 200t2 Combine like terms: 3600t + -3600t = 0 14900 + -3600t + 200t2 = 0 + -200t2 + 200t2 14900 + -3600t + 200t2 = -200t2 + 200t2 Combine like terms: -200t2 + 200t2 = 0 14900 + -3600t + 200t2 = 0 Factor out the Greatest Common Factor (GCF), '100'. 100(149 + -36t + 2t2) = 0 Ignore the factor 100.Subproblem 1
Set the factor '(149 + -36t + 2t2)' equal to zero and attempt to solve: Simplifying 149 + -36t + 2t2 = 0 Solving 149 + -36t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 74.5 + -18t + t2 = 0 Move the constant term to the right: Add '-74.5' to each side of the equation. 74.5 + -18t + -74.5 + t2 = 0 + -74.5 Reorder the terms: 74.5 + -74.5 + -18t + t2 = 0 + -74.5 Combine like terms: 74.5 + -74.5 = 0.0 0.0 + -18t + t2 = 0 + -74.5 -18t + t2 = 0 + -74.5 Combine like terms: 0 + -74.5 = -74.5 -18t + t2 = -74.5 The t term is -18t. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18t + 81 + t2 = -74.5 + 81 Reorder the terms: 81 + -18t + t2 = -74.5 + 81 Combine like terms: -74.5 + 81 = 6.5 81 + -18t + t2 = 6.5 Factor a perfect square on the left side: (t + -9)(t + -9) = 6.5 Calculate the square root of the right side: 2.549509757 Break this problem into two subproblems by setting (t + -9) equal to 2.549509757 and -2.549509757.Subproblem 1
t + -9 = 2.549509757 Simplifying t + -9 = 2.549509757 Reorder the terms: -9 + t = 2.549509757 Solving -9 + t = 2.549509757 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + t = 2.549509757 + 9 Combine like terms: -9 + 9 = 0 0 + t = 2.549509757 + 9 t = 2.549509757 + 9 Combine like terms: 2.549509757 + 9 = 11.549509757 t = 11.549509757 Simplifying t = 11.549509757Subproblem 2
t + -9 = -2.549509757 Simplifying t + -9 = -2.549509757 Reorder the terms: -9 + t = -2.549509757 Solving -9 + t = -2.549509757 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + t = -2.549509757 + 9 Combine like terms: -9 + 9 = 0 0 + t = -2.549509757 + 9 t = -2.549509757 + 9 Combine like terms: -2.549509757 + 9 = 6.450490243 t = 6.450490243 Simplifying t = 6.450490243Solution
The solution to the problem is based on the solutions from the subproblems. t = {11.549509757, 6.450490243}Solution
t = {11.549509757, 6.450490243}
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