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Simplifying 0 = 2 + 40t + -16t2 Solving 0 = 2 + 40t + -16t2 Solving for variable 't'. Combine like terms: 0 + -2 = -2 -2 + -40t + 16t2 = 2 + 40t + -16t2 + -2 + -40t + 16t2 Reorder the terms: -2 + -40t + 16t2 = 2 + -2 + 40t + -40t + -16t2 + 16t2 Combine like terms: 2 + -2 = 0 -2 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 -2 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 -2 + -40t + 16t2 = 0 + -16t2 + 16t2 -2 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -2 + -40t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + -20t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + -20t + 8t2)' equal to zero and attempt to solve: Simplifying -1 + -20t + 8t2 = 0 Solving -1 + -20t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + -2.5t + t2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -2.5t + 0.125 + t2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -2.5t + t2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -2.5t + t2 = 0 + 0.125 -2.5t + t2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -2.5t + t2 = 0.125 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = 0.125 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 0.125 + 1.5625 Combine like terms: 0.125 + 1.5625 = 1.6875 1.5625 + -2.5t + t2 = 1.6875 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 1.6875 Calculate the square root of the right side: 1.299038106 Break this problem into two subproblems by setting (t + -1.25) equal to 1.299038106 and -1.299038106.Subproblem 1
t + -1.25 = 1.299038106 Simplifying t + -1.25 = 1.299038106 Reorder the terms: -1.25 + t = 1.299038106 Solving -1.25 + t = 1.299038106 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 1.299038106 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.299038106 + 1.25 t = 1.299038106 + 1.25 Combine like terms: 1.299038106 + 1.25 = 2.549038106 t = 2.549038106 Simplifying t = 2.549038106Subproblem 2
t + -1.25 = -1.299038106 Simplifying t + -1.25 = -1.299038106 Reorder the terms: -1.25 + t = -1.299038106 Solving -1.25 + t = -1.299038106 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -1.299038106 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.299038106 + 1.25 t = -1.299038106 + 1.25 Combine like terms: -1.299038106 + 1.25 = -0.049038106 t = -0.049038106 Simplifying t = -0.049038106Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.549038106, -0.049038106}Solution
t = {2.549038106, -0.049038106}
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