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Simplifying 60(t) = -16t2 + 5t + 105 Reorder the terms: 60t = 105 + 5t + -16t2 Solving 60t = 105 + 5t + -16t2 Solving for variable 't'. Reorder the terms: -105 + 60t + -5t + 16t2 = 105 + 5t + -16t2 + -105 + -5t + 16t2 Combine like terms: 60t + -5t = 55t -105 + 55t + 16t2 = 105 + 5t + -16t2 + -105 + -5t + 16t2 Reorder the terms: -105 + 55t + 16t2 = 105 + -105 + 5t + -5t + -16t2 + 16t2 Combine like terms: 105 + -105 = 0 -105 + 55t + 16t2 = 0 + 5t + -5t + -16t2 + 16t2 -105 + 55t + 16t2 = 5t + -5t + -16t2 + 16t2 Combine like terms: 5t + -5t = 0 -105 + 55t + 16t2 = 0 + -16t2 + 16t2 -105 + 55t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -105 + 55t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -6.5625 + 3.4375t + t2 = 0 Move the constant term to the right: Add '6.5625' to each side of the equation. -6.5625 + 3.4375t + 6.5625 + t2 = 0 + 6.5625 Reorder the terms: -6.5625 + 6.5625 + 3.4375t + t2 = 0 + 6.5625 Combine like terms: -6.5625 + 6.5625 = 0.0000 0.0000 + 3.4375t + t2 = 0 + 6.5625 3.4375t + t2 = 0 + 6.5625 Combine like terms: 0 + 6.5625 = 6.5625 3.4375t + t2 = 6.5625 The t term is 3.4375t. Take half its coefficient (1.71875). Square it (2.954101563) and add it to both sides. Add '2.954101563' to each side of the equation. 3.4375t + 2.954101563 + t2 = 6.5625 + 2.954101563 Reorder the terms: 2.954101563 + 3.4375t + t2 = 6.5625 + 2.954101563 Combine like terms: 6.5625 + 2.954101563 = 9.516601563 2.954101563 + 3.4375t + t2 = 9.516601563 Factor a perfect square on the left side: (t + 1.71875)(t + 1.71875) = 9.516601563 Calculate the square root of the right side: 3.084898955 Break this problem into two subproblems by setting (t + 1.71875) equal to 3.084898955 and -3.084898955.Subproblem 1
t + 1.71875 = 3.084898955 Simplifying t + 1.71875 = 3.084898955 Reorder the terms: 1.71875 + t = 3.084898955 Solving 1.71875 + t = 3.084898955 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.71875' to each side of the equation. 1.71875 + -1.71875 + t = 3.084898955 + -1.71875 Combine like terms: 1.71875 + -1.71875 = 0.00000 0.00000 + t = 3.084898955 + -1.71875 t = 3.084898955 + -1.71875 Combine like terms: 3.084898955 + -1.71875 = 1.366148955 t = 1.366148955 Simplifying t = 1.366148955Subproblem 2
t + 1.71875 = -3.084898955 Simplifying t + 1.71875 = -3.084898955 Reorder the terms: 1.71875 + t = -3.084898955 Solving 1.71875 + t = -3.084898955 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.71875' to each side of the equation. 1.71875 + -1.71875 + t = -3.084898955 + -1.71875 Combine like terms: 1.71875 + -1.71875 = 0.00000 0.00000 + t = -3.084898955 + -1.71875 t = -3.084898955 + -1.71875 Combine like terms: -3.084898955 + -1.71875 = -4.803648955 t = -4.803648955 Simplifying t = -4.803648955Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.366148955, -4.803648955}
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