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Simplifying 45t = -16t2 + 33t + 45 Reorder the terms: 45t = 45 + 33t + -16t2 Solving 45t = 45 + 33t + -16t2 Solving for variable 't'. Reorder the terms: -45 + 45t + -33t + 16t2 = 45 + 33t + -16t2 + -45 + -33t + 16t2 Combine like terms: 45t + -33t = 12t -45 + 12t + 16t2 = 45 + 33t + -16t2 + -45 + -33t + 16t2 Reorder the terms: -45 + 12t + 16t2 = 45 + -45 + 33t + -33t + -16t2 + 16t2 Combine like terms: 45 + -45 = 0 -45 + 12t + 16t2 = 0 + 33t + -33t + -16t2 + 16t2 -45 + 12t + 16t2 = 33t + -33t + -16t2 + 16t2 Combine like terms: 33t + -33t = 0 -45 + 12t + 16t2 = 0 + -16t2 + 16t2 -45 + 12t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -45 + 12t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.8125 + 0.75t + t2 = 0 Move the constant term to the right: Add '2.8125' to each side of the equation. -2.8125 + 0.75t + 2.8125 + t2 = 0 + 2.8125 Reorder the terms: -2.8125 + 2.8125 + 0.75t + t2 = 0 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + 0.75t + t2 = 0 + 2.8125 0.75t + t2 = 0 + 2.8125 Combine like terms: 0 + 2.8125 = 2.8125 0.75t + t2 = 2.8125 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 2.8125 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 2.8125 + 0.140625 Combine like terms: 2.8125 + 0.140625 = 2.953125 0.140625 + 0.75t + t2 = 2.953125 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 2.953125 Calculate the square root of the right side: 1.718465886 Break this problem into two subproblems by setting (t + 0.375) equal to 1.718465886 and -1.718465886.Subproblem 1
t + 0.375 = 1.718465886 Simplifying t + 0.375 = 1.718465886 Reorder the terms: 0.375 + t = 1.718465886 Solving 0.375 + t = 1.718465886 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 1.718465886 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 1.718465886 + -0.375 t = 1.718465886 + -0.375 Combine like terms: 1.718465886 + -0.375 = 1.343465886 t = 1.343465886 Simplifying t = 1.343465886Subproblem 2
t + 0.375 = -1.718465886 Simplifying t + 0.375 = -1.718465886 Reorder the terms: 0.375 + t = -1.718465886 Solving 0.375 + t = -1.718465886 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -1.718465886 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -1.718465886 + -0.375 t = -1.718465886 + -0.375 Combine like terms: -1.718465886 + -0.375 = -2.093465886 t = -2.093465886 Simplifying t = -2.093465886Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.343465886, -2.093465886}
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