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Simplifying 8 = -16t2 + 40t + 3 Reorder the terms: 8 = 3 + 40t + -16t2 Solving 8 = 3 + 40t + -16t2 Solving for variable 't'. Combine like terms: 8 + -3 = 5 5 + -40t + 16t2 = 3 + 40t + -16t2 + -3 + -40t + 16t2 Reorder the terms: 5 + -40t + 16t2 = 3 + -3 + 40t + -40t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 5 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 5 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 5 + -40t + 16t2 = 0 + -16t2 + 16t2 5 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 5 + -40t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + -2.5t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + -2.5t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + -2.5t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + -2.5t + t2 = 0 + -0.3125 -2.5t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 -2.5t + t2 = -0.3125 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = -0.3125 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = -0.3125 + 1.5625 Combine like terms: -0.3125 + 1.5625 = 1.25 1.5625 + -2.5t + t2 = 1.25 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (t + -1.25) equal to 1.118033989 and -1.118033989.Subproblem 1
t + -1.25 = 1.118033989 Simplifying t + -1.25 = 1.118033989 Reorder the terms: -1.25 + t = 1.118033989 Solving -1.25 + t = 1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 1.118033989 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.118033989 + 1.25 t = 1.118033989 + 1.25 Combine like terms: 1.118033989 + 1.25 = 2.368033989 t = 2.368033989 Simplifying t = 2.368033989Subproblem 2
t + -1.25 = -1.118033989 Simplifying t + -1.25 = -1.118033989 Reorder the terms: -1.25 + t = -1.118033989 Solving -1.25 + t = -1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -1.118033989 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.118033989 + 1.25 t = -1.118033989 + 1.25 Combine like terms: -1.118033989 + 1.25 = 0.131966011 t = 0.131966011 Simplifying t = 0.131966011Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.368033989, 0.131966011}
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