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Simplifying 12 = -2.7t2 + 30t + 6.5 Reorder the terms: 12 = 6.5 + 30t + -2.7t2 Solving 12 = 6.5 + 30t + -2.7t2 Solving for variable 't'. Combine like terms: 12 + -6.5 = 5.5 5.5 + -30t + 2.7t2 = 6.5 + 30t + -2.7t2 + -6.5 + -30t + 2.7t2 Reorder the terms: 5.5 + -30t + 2.7t2 = 6.5 + -6.5 + 30t + -30t + -2.7t2 + 2.7t2 Combine like terms: 6.5 + -6.5 = 0.0 5.5 + -30t + 2.7t2 = 0.0 + 30t + -30t + -2.7t2 + 2.7t2 5.5 + -30t + 2.7t2 = 30t + -30t + -2.7t2 + 2.7t2 Combine like terms: 30t + -30t = 0 5.5 + -30t + 2.7t2 = 0 + -2.7t2 + 2.7t2 5.5 + -30t + 2.7t2 = -2.7t2 + 2.7t2 Combine like terms: -2.7t2 + 2.7t2 = 0.0 5.5 + -30t + 2.7t2 = 0.0 Begin completing the square. Divide all terms by 2.7 the coefficient of the squared term: Divide each side by '2.7'. 2.037037037 + -11.11111111t + t2 = 0 Move the constant term to the right: Add '-2.037037037' to each side of the equation. 2.037037037 + -11.11111111t + -2.037037037 + t2 = 0 + -2.037037037 Reorder the terms: 2.037037037 + -2.037037037 + -11.11111111t + t2 = 0 + -2.037037037 Combine like terms: 2.037037037 + -2.037037037 = 0.000000000 0.000000000 + -11.11111111t + t2 = 0 + -2.037037037 -11.11111111t + t2 = 0 + -2.037037037 Combine like terms: 0 + -2.037037037 = -2.037037037 -11.11111111t + t2 = -2.037037037 The t term is -11.11111111t. Take half its coefficient (-5.555555555). Square it (30.86419752) and add it to both sides. Add '30.86419752' to each side of the equation. -11.11111111t + 30.86419752 + t2 = -2.037037037 + 30.86419752 Reorder the terms: 30.86419752 + -11.11111111t + t2 = -2.037037037 + 30.86419752 Combine like terms: -2.037037037 + 30.86419752 = 28.827160483 30.86419752 + -11.11111111t + t2 = 28.827160483 Factor a perfect square on the left side: (t + -5.555555555)(t + -5.555555555) = 28.827160483 Calculate the square root of the right side: 5.369093078 Break this problem into two subproblems by setting (t + -5.555555555) equal to 5.369093078 and -5.369093078.Subproblem 1
t + -5.555555555 = 5.369093078 Simplifying t + -5.555555555 = 5.369093078 Reorder the terms: -5.555555555 + t = 5.369093078 Solving -5.555555555 + t = 5.369093078 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.555555555' to each side of the equation. -5.555555555 + 5.555555555 + t = 5.369093078 + 5.555555555 Combine like terms: -5.555555555 + 5.555555555 = 0.000000000 0.000000000 + t = 5.369093078 + 5.555555555 t = 5.369093078 + 5.555555555 Combine like terms: 5.369093078 + 5.555555555 = 10.924648633 t = 10.924648633 Simplifying t = 10.924648633Subproblem 2
t + -5.555555555 = -5.369093078 Simplifying t + -5.555555555 = -5.369093078 Reorder the terms: -5.555555555 + t = -5.369093078 Solving -5.555555555 + t = -5.369093078 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.555555555' to each side of the equation. -5.555555555 + 5.555555555 + t = -5.369093078 + 5.555555555 Combine like terms: -5.555555555 + 5.555555555 = 0.000000000 0.000000000 + t = -5.369093078 + 5.555555555 t = -5.369093078 + 5.555555555 Combine like terms: -5.369093078 + 5.555555555 = 0.186462477 t = 0.186462477 Simplifying t = 0.186462477Solution
The solution to the problem is based on the solutions from the subproblems. t = {10.924648633, 0.186462477}
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