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Simplifying 0 = -16t2 + 38t + 7 Reorder the terms: 0 = 7 + 38t + -16t2 Solving 0 = 7 + 38t + -16t2 Solving for variable 't'. Combine like terms: 0 + -7 = -7 -7 + -38t + 16t2 = 7 + 38t + -16t2 + -7 + -38t + 16t2 Reorder the terms: -7 + -38t + 16t2 = 7 + -7 + 38t + -38t + -16t2 + 16t2 Combine like terms: 7 + -7 = 0 -7 + -38t + 16t2 = 0 + 38t + -38t + -16t2 + 16t2 -7 + -38t + 16t2 = 38t + -38t + -16t2 + 16t2 Combine like terms: 38t + -38t = 0 -7 + -38t + 16t2 = 0 + -16t2 + 16t2 -7 + -38t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -7 + -38t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.4375 + -2.375t + t2 = 0 Move the constant term to the right: Add '0.4375' to each side of the equation. -0.4375 + -2.375t + 0.4375 + t2 = 0 + 0.4375 Reorder the terms: -0.4375 + 0.4375 + -2.375t + t2 = 0 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + -2.375t + t2 = 0 + 0.4375 -2.375t + t2 = 0 + 0.4375 Combine like terms: 0 + 0.4375 = 0.4375 -2.375t + t2 = 0.4375 The t term is -2.375t. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375t + 1.41015625 + t2 = 0.4375 + 1.41015625 Reorder the terms: 1.41015625 + -2.375t + t2 = 0.4375 + 1.41015625 Combine like terms: 0.4375 + 1.41015625 = 1.84765625 1.41015625 + -2.375t + t2 = 1.84765625 Factor a perfect square on the left side: (t + -1.1875)(t + -1.1875) = 1.84765625 Calculate the square root of the right side: 1.359285198 Break this problem into two subproblems by setting (t + -1.1875) equal to 1.359285198 and -1.359285198.Subproblem 1
t + -1.1875 = 1.359285198 Simplifying t + -1.1875 = 1.359285198 Reorder the terms: -1.1875 + t = 1.359285198 Solving -1.1875 + t = 1.359285198 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = 1.359285198 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = 1.359285198 + 1.1875 t = 1.359285198 + 1.1875 Combine like terms: 1.359285198 + 1.1875 = 2.546785198 t = 2.546785198 Simplifying t = 2.546785198Subproblem 2
t + -1.1875 = -1.359285198 Simplifying t + -1.1875 = -1.359285198 Reorder the terms: -1.1875 + t = -1.359285198 Solving -1.1875 + t = -1.359285198 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = -1.359285198 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = -1.359285198 + 1.1875 t = -1.359285198 + 1.1875 Combine like terms: -1.359285198 + 1.1875 = -0.171785198 t = -0.171785198 Simplifying t = -0.171785198Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.546785198, -0.171785198}
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