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Simplifying 5 = -16t2 + 128t Reorder the terms: 5 = 128t + -16t2 Solving 5 = 128t + -16t2 Solving for variable 't'. Reorder the terms: 5 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 5 + -128t + 16t2 = 0 + -16t2 + 16t2 5 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 5 + -128t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + -8t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + -8t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + -8t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + -8t + t2 = 0 + -0.3125 -8t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 -8t + t2 = -0.3125 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = -0.3125 + 16 Reorder the terms: 16 + -8t + t2 = -0.3125 + 16 Combine like terms: -0.3125 + 16 = 15.6875 16 + -8t + t2 = 15.6875 Factor a perfect square on the left side: (t + -4)(t + -4) = 15.6875 Calculate the square root of the right side: 3.960744879 Break this problem into two subproblems by setting (t + -4) equal to 3.960744879 and -3.960744879.Subproblem 1
t + -4 = 3.960744879 Simplifying t + -4 = 3.960744879 Reorder the terms: -4 + t = 3.960744879 Solving -4 + t = 3.960744879 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 3.960744879 + 4 Combine like terms: -4 + 4 = 0 0 + t = 3.960744879 + 4 t = 3.960744879 + 4 Combine like terms: 3.960744879 + 4 = 7.960744879 t = 7.960744879 Simplifying t = 7.960744879Subproblem 2
t + -4 = -3.960744879 Simplifying t + -4 = -3.960744879 Reorder the terms: -4 + t = -3.960744879 Solving -4 + t = -3.960744879 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -3.960744879 + 4 Combine like terms: -4 + 4 = 0 0 + t = -3.960744879 + 4 t = -3.960744879 + 4 Combine like terms: -3.960744879 + 4 = 0.039255121 t = 0.039255121 Simplifying t = 0.039255121Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.960744879, 0.039255121}
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