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Simplifying 12 = 4 + 23t + -16t2 Solving 12 = 4 + 23t + -16t2 Solving for variable 't'. Combine like terms: 12 + -4 = 8 8 + -23t + 16t2 = 4 + 23t + -16t2 + -4 + -23t + 16t2 Reorder the terms: 8 + -23t + 16t2 = 4 + -4 + 23t + -23t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 8 + -23t + 16t2 = 0 + 23t + -23t + -16t2 + 16t2 8 + -23t + 16t2 = 23t + -23t + -16t2 + 16t2 Combine like terms: 23t + -23t = 0 8 + -23t + 16t2 = 0 + -16t2 + 16t2 8 + -23t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 8 + -23t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.5 + -1.4375t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.4375t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.4375t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.4375t + t2 = 0 + -0.5 -1.4375t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -1.4375t + t2 = -0.5 The t term is -1.4375t. Take half its coefficient (-0.71875). Square it (0.5166015625) and add it to both sides. Add '0.5166015625' to each side of the equation. -1.4375t + 0.5166015625 + t2 = -0.5 + 0.5166015625 Reorder the terms: 0.5166015625 + -1.4375t + t2 = -0.5 + 0.5166015625 Combine like terms: -0.5 + 0.5166015625 = 0.0166015625 0.5166015625 + -1.4375t + t2 = 0.0166015625 Factor a perfect square on the left side: (t + -0.71875)(t + -0.71875) = 0.0166015625 Calculate the square root of the right side: 0.128847051 Break this problem into two subproblems by setting (t + -0.71875) equal to 0.128847051 and -0.128847051.Subproblem 1
t + -0.71875 = 0.128847051 Simplifying t + -0.71875 = 0.128847051 Reorder the terms: -0.71875 + t = 0.128847051 Solving -0.71875 + t = 0.128847051 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + t = 0.128847051 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = 0.128847051 + 0.71875 t = 0.128847051 + 0.71875 Combine like terms: 0.128847051 + 0.71875 = 0.847597051 t = 0.847597051 Simplifying t = 0.847597051Subproblem 2
t + -0.71875 = -0.128847051 Simplifying t + -0.71875 = -0.128847051 Reorder the terms: -0.71875 + t = -0.128847051 Solving -0.71875 + t = -0.128847051 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + t = -0.128847051 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = -0.128847051 + 0.71875 t = -0.128847051 + 0.71875 Combine like terms: -0.128847051 + 0.71875 = 0.589902949 t = 0.589902949 Simplifying t = 0.589902949Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.847597051, 0.589902949}
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