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Simplifying 5(2t + -5) * 4t = 50 + -1t Reorder the terms: 5(-5 + 2t) * 4t = 50 + -1t Reorder the terms for easier multiplication: 5 * 4t(-5 + 2t) = 50 + -1t Multiply 5 * 4 20t(-5 + 2t) = 50 + -1t (-5 * 20t + 2t * 20t) = 50 + -1t (-100t + 40t2) = 50 + -1t Solving -100t + 40t2 = 50 + -1t Solving for variable 't'. Reorder the terms: -50 + -100t + t + 40t2 = 50 + -1t + -50 + t Combine like terms: -100t + t = -99t -50 + -99t + 40t2 = 50 + -1t + -50 + t Reorder the terms: -50 + -99t + 40t2 = 50 + -50 + -1t + t Combine like terms: 50 + -50 = 0 -50 + -99t + 40t2 = 0 + -1t + t -50 + -99t + 40t2 = -1t + t Combine like terms: -1t + t = 0 -50 + -99t + 40t2 = 0 Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -1.25 + -2.475t + t2 = 0.0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -2.475t + 1.25 + t2 = 0.0 + 1.25 Reorder the terms: -1.25 + 1.25 + -2.475t + t2 = 0.0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -2.475t + t2 = 0.0 + 1.25 -2.475t + t2 = 0.0 + 1.25 Combine like terms: 0.0 + 1.25 = 1.25 -2.475t + t2 = 1.25 The t term is -2.475t. Take half its coefficient (-1.2375). Square it (1.53140625) and add it to both sides. Add '1.53140625' to each side of the equation. -2.475t + 1.53140625 + t2 = 1.25 + 1.53140625 Reorder the terms: 1.53140625 + -2.475t + t2 = 1.25 + 1.53140625 Combine like terms: 1.25 + 1.53140625 = 2.78140625 1.53140625 + -2.475t + t2 = 2.78140625 Factor a perfect square on the left side: (t + -1.2375)(t + -1.2375) = 2.78140625 Calculate the square root of the right side: 1.667754853 Break this problem into two subproblems by setting (t + -1.2375) equal to 1.667754853 and -1.667754853.Subproblem 1
t + -1.2375 = 1.667754853 Simplifying t + -1.2375 = 1.667754853 Reorder the terms: -1.2375 + t = 1.667754853 Solving -1.2375 + t = 1.667754853 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.2375' to each side of the equation. -1.2375 + 1.2375 + t = 1.667754853 + 1.2375 Combine like terms: -1.2375 + 1.2375 = 0.0000 0.0000 + t = 1.667754853 + 1.2375 t = 1.667754853 + 1.2375 Combine like terms: 1.667754853 + 1.2375 = 2.905254853 t = 2.905254853 Simplifying t = 2.905254853Subproblem 2
t + -1.2375 = -1.667754853 Simplifying t + -1.2375 = -1.667754853 Reorder the terms: -1.2375 + t = -1.667754853 Solving -1.2375 + t = -1.667754853 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.2375' to each side of the equation. -1.2375 + 1.2375 + t = -1.667754853 + 1.2375 Combine like terms: -1.2375 + 1.2375 = 0.0000 0.0000 + t = -1.667754853 + 1.2375 t = -1.667754853 + 1.2375 Combine like terms: -1.667754853 + 1.2375 = -0.430254853 t = -0.430254853 Simplifying t = -0.430254853Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.905254853, -0.430254853}
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