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Simplifying t = -16t2 + 100t + 165 Reorder the terms: t = 165 + 100t + -16t2 Solving t = 165 + 100t + -16t2 Solving for variable 't'. Reorder the terms: -165 + t + -100t + 16t2 = 165 + 100t + -16t2 + -165 + -100t + 16t2 Combine like terms: t + -100t = -99t -165 + -99t + 16t2 = 165 + 100t + -16t2 + -165 + -100t + 16t2 Reorder the terms: -165 + -99t + 16t2 = 165 + -165 + 100t + -100t + -16t2 + 16t2 Combine like terms: 165 + -165 = 0 -165 + -99t + 16t2 = 0 + 100t + -100t + -16t2 + 16t2 -165 + -99t + 16t2 = 100t + -100t + -16t2 + 16t2 Combine like terms: 100t + -100t = 0 -165 + -99t + 16t2 = 0 + -16t2 + 16t2 -165 + -99t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -165 + -99t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -10.3125 + -6.1875t + t2 = 0 Move the constant term to the right: Add '10.3125' to each side of the equation. -10.3125 + -6.1875t + 10.3125 + t2 = 0 + 10.3125 Reorder the terms: -10.3125 + 10.3125 + -6.1875t + t2 = 0 + 10.3125 Combine like terms: -10.3125 + 10.3125 = 0.0000 0.0000 + -6.1875t + t2 = 0 + 10.3125 -6.1875t + t2 = 0 + 10.3125 Combine like terms: 0 + 10.3125 = 10.3125 -6.1875t + t2 = 10.3125 The t term is -6.1875t. Take half its coefficient (-3.09375). Square it (9.571289063) and add it to both sides. Add '9.571289063' to each side of the equation. -6.1875t + 9.571289063 + t2 = 10.3125 + 9.571289063 Reorder the terms: 9.571289063 + -6.1875t + t2 = 10.3125 + 9.571289063 Combine like terms: 10.3125 + 9.571289063 = 19.883789063 9.571289063 + -6.1875t + t2 = 19.883789063 Factor a perfect square on the left side: (t + -3.09375)(t + -3.09375) = 19.883789063 Calculate the square root of the right side: 4.459124248 Break this problem into two subproblems by setting (t + -3.09375) equal to 4.459124248 and -4.459124248.Subproblem 1
t + -3.09375 = 4.459124248 Simplifying t + -3.09375 = 4.459124248 Reorder the terms: -3.09375 + t = 4.459124248 Solving -3.09375 + t = 4.459124248 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.09375' to each side of the equation. -3.09375 + 3.09375 + t = 4.459124248 + 3.09375 Combine like terms: -3.09375 + 3.09375 = 0.00000 0.00000 + t = 4.459124248 + 3.09375 t = 4.459124248 + 3.09375 Combine like terms: 4.459124248 + 3.09375 = 7.552874248 t = 7.552874248 Simplifying t = 7.552874248Subproblem 2
t + -3.09375 = -4.459124248 Simplifying t + -3.09375 = -4.459124248 Reorder the terms: -3.09375 + t = -4.459124248 Solving -3.09375 + t = -4.459124248 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.09375' to each side of the equation. -3.09375 + 3.09375 + t = -4.459124248 + 3.09375 Combine like terms: -3.09375 + 3.09375 = 0.00000 0.00000 + t = -4.459124248 + 3.09375 t = -4.459124248 + 3.09375 Combine like terms: -4.459124248 + 3.09375 = -1.365374248 t = -1.365374248 Simplifying t = -1.365374248Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.552874248, -1.365374248}
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