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Simplifying t = 217 + -17t + -16t2 Solving t = 217 + -17t + -16t2 Solving for variable 't'. Reorder the terms: -217 + t + 17t + 16t2 = 217 + -17t + -16t2 + -217 + 17t + 16t2 Combine like terms: t + 17t = 18t -217 + 18t + 16t2 = 217 + -17t + -16t2 + -217 + 17t + 16t2 Reorder the terms: -217 + 18t + 16t2 = 217 + -217 + -17t + 17t + -16t2 + 16t2 Combine like terms: 217 + -217 = 0 -217 + 18t + 16t2 = 0 + -17t + 17t + -16t2 + 16t2 -217 + 18t + 16t2 = -17t + 17t + -16t2 + 16t2 Combine like terms: -17t + 17t = 0 -217 + 18t + 16t2 = 0 + -16t2 + 16t2 -217 + 18t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -217 + 18t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -13.5625 + 1.125t + t2 = 0 Move the constant term to the right: Add '13.5625' to each side of the equation. -13.5625 + 1.125t + 13.5625 + t2 = 0 + 13.5625 Reorder the terms: -13.5625 + 13.5625 + 1.125t + t2 = 0 + 13.5625 Combine like terms: -13.5625 + 13.5625 = 0.0000 0.0000 + 1.125t + t2 = 0 + 13.5625 1.125t + t2 = 0 + 13.5625 Combine like terms: 0 + 13.5625 = 13.5625 1.125t + t2 = 13.5625 The t term is 1.125t. Take half its coefficient (0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. 1.125t + 0.31640625 + t2 = 13.5625 + 0.31640625 Reorder the terms: 0.31640625 + 1.125t + t2 = 13.5625 + 0.31640625 Combine like terms: 13.5625 + 0.31640625 = 13.87890625 0.31640625 + 1.125t + t2 = 13.87890625 Factor a perfect square on the left side: (t + 0.5625)(t + 0.5625) = 13.87890625 Calculate the square root of the right side: 3.72544041 Break this problem into two subproblems by setting (t + 0.5625) equal to 3.72544041 and -3.72544041.Subproblem 1
t + 0.5625 = 3.72544041 Simplifying t + 0.5625 = 3.72544041 Reorder the terms: 0.5625 + t = 3.72544041 Solving 0.5625 + t = 3.72544041 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + t = 3.72544041 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + t = 3.72544041 + -0.5625 t = 3.72544041 + -0.5625 Combine like terms: 3.72544041 + -0.5625 = 3.16294041 t = 3.16294041 Simplifying t = 3.16294041Subproblem 2
t + 0.5625 = -3.72544041 Simplifying t + 0.5625 = -3.72544041 Reorder the terms: 0.5625 + t = -3.72544041 Solving 0.5625 + t = -3.72544041 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + t = -3.72544041 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + t = -3.72544041 + -0.5625 t = -3.72544041 + -0.5625 Combine like terms: -3.72544041 + -0.5625 = -4.28794041 t = -4.28794041 Simplifying t = -4.28794041Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.16294041, -4.28794041}
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