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Simplifying 2 = -16t2 + 38t + 6 Reorder the terms: 2 = 6 + 38t + -16t2 Solving 2 = 6 + 38t + -16t2 Solving for variable 't'. Combine like terms: 2 + -6 = -4 -4 + -38t + 16t2 = 6 + 38t + -16t2 + -6 + -38t + 16t2 Reorder the terms: -4 + -38t + 16t2 = 6 + -6 + 38t + -38t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 -4 + -38t + 16t2 = 0 + 38t + -38t + -16t2 + 16t2 -4 + -38t + 16t2 = 38t + -38t + -16t2 + 16t2 Combine like terms: 38t + -38t = 0 -4 + -38t + 16t2 = 0 + -16t2 + 16t2 -4 + -38t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -4 + -38t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-2 + -19t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2 + -19t + 8t2)' equal to zero and attempt to solve: Simplifying -2 + -19t + 8t2 = 0 Solving -2 + -19t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.25 + -2.375t + t2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -2.375t + 0.25 + t2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -2.375t + t2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -2.375t + t2 = 0 + 0.25 -2.375t + t2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -2.375t + t2 = 0.25 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 = 0.25 + 1.41015625 Reorder the terms: 1.41015625 + -2.375t + t2 = 0.25 + 1.41015625 Combine like terms: 0.25 + 1.41015625 = 1.66015625 1.41015625 + -2.375t + t2 = 1.66015625 Factor a perfect square on the left side: (t + -1.1875)(t + -1.1875) = 1.66015625 Calculate the square root of the right side: 1.288470508 Break this problem into two subproblems by setting (t + -1.1875) equal to 1.288470508 and -1.288470508.Subproblem 1
t + -1.1875 = 1.288470508 Simplifying t + -1.1875 = 1.288470508 Reorder the terms: -1.1875 + t = 1.288470508 Solving -1.1875 + t = 1.288470508 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 = 1.288470508 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = 1.288470508 + 1.1875 t = 1.288470508 + 1.1875 Combine like terms: 1.288470508 + 1.1875 = 2.475970508 t = 2.475970508 Simplifying t = 2.475970508Subproblem 2
t + -1.1875 = -1.288470508 Simplifying t + -1.1875 = -1.288470508 Reorder the terms: -1.1875 + t = -1.288470508 Solving -1.1875 + t = -1.288470508 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 = -1.288470508 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = -1.288470508 + 1.1875 t = -1.288470508 + 1.1875 Combine like terms: -1.288470508 + 1.1875 = -0.100970508 t = -0.100970508 Simplifying t = -0.100970508Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.475970508, -0.100970508}Solution
t = {2.475970508, -0.100970508}
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