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Simplifying -16t2 + -30t + -1 = 0 Reorder the terms: -1 + -30t + -16t2 = 0 Solving -1 + -30t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 30t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 30t + 16t2)' equal to zero and attempt to solve: Simplifying 1 + 30t + 16t2 = 0 Solving 1 + 30t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.0625 + 1.875t + t2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + 1.875t + -0.0625 + t2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + 1.875t + t2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + 1.875t + t2 = 0 + -0.0625 1.875t + t2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 1.875t + t2 = -0.0625 The t term is 1.875t. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875t + 0.87890625 + t2 = -0.0625 + 0.87890625 Reorder the terms: 0.87890625 + 1.875t + t2 = -0.0625 + 0.87890625 Combine like terms: -0.0625 + 0.87890625 = 0.81640625 0.87890625 + 1.875t + t2 = 0.81640625 Factor a perfect square on the left side: (t + 0.9375)(t + 0.9375) = 0.81640625 Calculate the square root of the right side: 0.903552018 Break this problem into two subproblems by setting (t + 0.9375) equal to 0.903552018 and -0.903552018.Subproblem 1
t + 0.9375 = 0.903552018 Simplifying t + 0.9375 = 0.903552018 Reorder the terms: 0.9375 + t = 0.903552018 Solving 0.9375 + t = 0.903552018 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + t = 0.903552018 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + t = 0.903552018 + -0.9375 t = 0.903552018 + -0.9375 Combine like terms: 0.903552018 + -0.9375 = -0.033947982 t = -0.033947982 Simplifying t = -0.033947982Subproblem 2
t + 0.9375 = -0.903552018 Simplifying t + 0.9375 = -0.903552018 Reorder the terms: 0.9375 + t = -0.903552018 Solving 0.9375 + t = -0.903552018 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + t = -0.903552018 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + t = -0.903552018 + -0.9375 t = -0.903552018 + -0.9375 Combine like terms: -0.903552018 + -0.9375 = -1.841052018 t = -1.841052018 Simplifying t = -1.841052018Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.033947982, -1.841052018}Solution
t = {-0.033947982, -1.841052018}
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