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Simplifying 0 = 16t2 + 38t + 5 Reorder the terms: 0 = 5 + 38t + 16t2 Solving 0 = 5 + 38t + 16t2 Solving for variable 't'. Combine like terms: 0 + -5 = -5 -5 + -38t + -16t2 = 5 + 38t + 16t2 + -5 + -38t + -16t2 Reorder the terms: -5 + -38t + -16t2 = 5 + -5 + 38t + -38t + 16t2 + -16t2 Combine like terms: 5 + -5 = 0 -5 + -38t + -16t2 = 0 + 38t + -38t + 16t2 + -16t2 -5 + -38t + -16t2 = 38t + -38t + 16t2 + -16t2 Combine like terms: 38t + -38t = 0 -5 + -38t + -16t2 = 0 + 16t2 + -16t2 -5 + -38t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -5 + -38t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 38t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 38t + 16t2)' equal to zero and attempt to solve: Simplifying 5 + 38t + 16t2 = 0 Solving 5 + 38t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + 2.375t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + 2.375t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + 2.375t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + 2.375t + t2 = 0 + -0.3125 2.375t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 2.375t + t2 = -0.3125 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 = -0.3125 + 1.41015625 Reorder the terms: 1.41015625 + 2.375t + t2 = -0.3125 + 1.41015625 Combine like terms: -0.3125 + 1.41015625 = 1.09765625 1.41015625 + 2.375t + t2 = 1.09765625 Factor a perfect square on the left side: (t + 1.1875)(t + 1.1875) = 1.09765625 Calculate the square root of the right side: 1.047690913 Break this problem into two subproblems by setting (t + 1.1875) equal to 1.047690913 and -1.047690913.Subproblem 1
t + 1.1875 = 1.047690913 Simplifying t + 1.1875 = 1.047690913 Reorder the terms: 1.1875 + t = 1.047690913 Solving 1.1875 + t = 1.047690913 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 = 1.047690913 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + t = 1.047690913 + -1.1875 t = 1.047690913 + -1.1875 Combine like terms: 1.047690913 + -1.1875 = -0.139809087 t = -0.139809087 Simplifying t = -0.139809087Subproblem 2
t + 1.1875 = -1.047690913 Simplifying t + 1.1875 = -1.047690913 Reorder the terms: 1.1875 + t = -1.047690913 Solving 1.1875 + t = -1.047690913 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 = -1.047690913 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + t = -1.047690913 + -1.1875 t = -1.047690913 + -1.1875 Combine like terms: -1.047690913 + -1.1875 = -2.235190913 t = -2.235190913 Simplifying t = -2.235190913Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.139809087, -2.235190913}Solution
t = {-0.139809087, -2.235190913}
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