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Simplifying 3 = -16t2 + 48t Reorder the terms: 3 = 48t + -16t2 Solving 3 = 48t + -16t2 Solving for variable 't'. Reorder the terms: 3 + -48t + 16t2 = 48t + -48t + -16t2 + 16t2 Combine like terms: 48t + -48t = 0 3 + -48t + 16t2 = 0 + -16t2 + 16t2 3 + -48t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 3 + -48t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + -3t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -3t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -3t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -3t + t2 = 0 + -0.1875 -3t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -3t + t2 = -0.1875 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = -0.1875 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -0.1875 + 2.25 Combine like terms: -0.1875 + 2.25 = 2.0625 2.25 + -3t + t2 = 2.0625 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (t + -1.5) equal to 1.436140662 and -1.436140662.Subproblem 1
t + -1.5 = 1.436140662 Simplifying t + -1.5 = 1.436140662 Reorder the terms: -1.5 + t = 1.436140662 Solving -1.5 + t = 1.436140662 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 1.436140662 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.436140662 + 1.5 t = 1.436140662 + 1.5 Combine like terms: 1.436140662 + 1.5 = 2.936140662 t = 2.936140662 Simplifying t = 2.936140662Subproblem 2
t + -1.5 = -1.436140662 Simplifying t + -1.5 = -1.436140662 Reorder the terms: -1.5 + t = -1.436140662 Solving -1.5 + t = -1.436140662 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -1.436140662 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.436140662 + 1.5 t = -1.436140662 + 1.5 Combine like terms: -1.436140662 + 1.5 = 0.063859338 t = 0.063859338 Simplifying t = 0.063859338Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.936140662, 0.063859338}
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