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Simplifying t = -16t2 + 193t + 45 Reorder the terms: t = 45 + 193t + -16t2 Solving t = 45 + 193t + -16t2 Solving for variable 't'. Reorder the terms: -45 + t + -193t + 16t2 = 45 + 193t + -16t2 + -45 + -193t + 16t2 Combine like terms: t + -193t = -192t -45 + -192t + 16t2 = 45 + 193t + -16t2 + -45 + -193t + 16t2 Reorder the terms: -45 + -192t + 16t2 = 45 + -45 + 193t + -193t + -16t2 + 16t2 Combine like terms: 45 + -45 = 0 -45 + -192t + 16t2 = 0 + 193t + -193t + -16t2 + 16t2 -45 + -192t + 16t2 = 193t + -193t + -16t2 + 16t2 Combine like terms: 193t + -193t = 0 -45 + -192t + 16t2 = 0 + -16t2 + 16t2 -45 + -192t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -45 + -192t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.8125 + -12t + t2 = 0 Move the constant term to the right: Add '2.8125' to each side of the equation. -2.8125 + -12t + 2.8125 + t2 = 0 + 2.8125 Reorder the terms: -2.8125 + 2.8125 + -12t + t2 = 0 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + -12t + t2 = 0 + 2.8125 -12t + t2 = 0 + 2.8125 Combine like terms: 0 + 2.8125 = 2.8125 -12t + t2 = 2.8125 The t term is -12t. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12t + 36 + t2 = 2.8125 + 36 Reorder the terms: 36 + -12t + t2 = 2.8125 + 36 Combine like terms: 2.8125 + 36 = 38.8125 36 + -12t + t2 = 38.8125 Factor a perfect square on the left side: (t + -6)(t + -6) = 38.8125 Calculate the square root of the right side: 6.229967897 Break this problem into two subproblems by setting (t + -6) equal to 6.229967897 and -6.229967897.Subproblem 1
t + -6 = 6.229967897 Simplifying t + -6 = 6.229967897 Reorder the terms: -6 + t = 6.229967897 Solving -6 + t = 6.229967897 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = 6.229967897 + 6 Combine like terms: -6 + 6 = 0 0 + t = 6.229967897 + 6 t = 6.229967897 + 6 Combine like terms: 6.229967897 + 6 = 12.229967897 t = 12.229967897 Simplifying t = 12.229967897Subproblem 2
t + -6 = -6.229967897 Simplifying t + -6 = -6.229967897 Reorder the terms: -6 + t = -6.229967897 Solving -6 + t = -6.229967897 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = -6.229967897 + 6 Combine like terms: -6 + 6 = 0 0 + t = -6.229967897 + 6 t = -6.229967897 + 6 Combine like terms: -6.229967897 + 6 = -0.229967897 t = -0.229967897 Simplifying t = -0.229967897Solution
The solution to the problem is based on the solutions from the subproblems. t = {12.229967897, -0.229967897}
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