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Simplifying 25t = -16t2 + 8t + 24 Reorder the terms: 25t = 24 + 8t + -16t2 Solving 25t = 24 + 8t + -16t2 Solving for variable 't'. Reorder the terms: -24 + 25t + -8t + 16t2 = 24 + 8t + -16t2 + -24 + -8t + 16t2 Combine like terms: 25t + -8t = 17t -24 + 17t + 16t2 = 24 + 8t + -16t2 + -24 + -8t + 16t2 Reorder the terms: -24 + 17t + 16t2 = 24 + -24 + 8t + -8t + -16t2 + 16t2 Combine like terms: 24 + -24 = 0 -24 + 17t + 16t2 = 0 + 8t + -8t + -16t2 + 16t2 -24 + 17t + 16t2 = 8t + -8t + -16t2 + 16t2 Combine like terms: 8t + -8t = 0 -24 + 17t + 16t2 = 0 + -16t2 + 16t2 -24 + 17t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -24 + 17t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.5 + 1.0625t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.0625t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.0625t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.0625t + t2 = 0 + 1.5 1.0625t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.0625t + t2 = 1.5 The t term is 1.0625t. Take half its coefficient (0.53125). Square it (0.2822265625) and add it to both sides. Add '0.2822265625' to each side of the equation. 1.0625t + 0.2822265625 + t2 = 1.5 + 0.2822265625 Reorder the terms: 0.2822265625 + 1.0625t + t2 = 1.5 + 0.2822265625 Combine like terms: 1.5 + 0.2822265625 = 1.7822265625 0.2822265625 + 1.0625t + t2 = 1.7822265625 Factor a perfect square on the left side: (t + 0.53125)(t + 0.53125) = 1.7822265625 Calculate the square root of the right side: 1.335000585 Break this problem into two subproblems by setting (t + 0.53125) equal to 1.335000585 and -1.335000585.Subproblem 1
t + 0.53125 = 1.335000585 Simplifying t + 0.53125 = 1.335000585 Reorder the terms: 0.53125 + t = 1.335000585 Solving 0.53125 + t = 1.335000585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.53125' to each side of the equation. 0.53125 + -0.53125 + t = 1.335000585 + -0.53125 Combine like terms: 0.53125 + -0.53125 = 0.00000 0.00000 + t = 1.335000585 + -0.53125 t = 1.335000585 + -0.53125 Combine like terms: 1.335000585 + -0.53125 = 0.803750585 t = 0.803750585 Simplifying t = 0.803750585Subproblem 2
t + 0.53125 = -1.335000585 Simplifying t + 0.53125 = -1.335000585 Reorder the terms: 0.53125 + t = -1.335000585 Solving 0.53125 + t = -1.335000585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.53125' to each side of the equation. 0.53125 + -0.53125 + t = -1.335000585 + -0.53125 Combine like terms: 0.53125 + -0.53125 = 0.00000 0.00000 + t = -1.335000585 + -0.53125 t = -1.335000585 + -0.53125 Combine like terms: -1.335000585 + -0.53125 = -1.866250585 t = -1.866250585 Simplifying t = -1.866250585Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.803750585, -1.866250585}
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