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Simplifying t = -16t2 + 32t + 5 Reorder the terms: t = 5 + 32t + -16t2 Solving t = 5 + 32t + -16t2 Solving for variable 't'. Reorder the terms: -5 + t + -32t + 16t2 = 5 + 32t + -16t2 + -5 + -32t + 16t2 Combine like terms: t + -32t = -31t -5 + -31t + 16t2 = 5 + 32t + -16t2 + -5 + -32t + 16t2 Reorder the terms: -5 + -31t + 16t2 = 5 + -5 + 32t + -32t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -31t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -5 + -31t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -5 + -31t + 16t2 = 0 + -16t2 + 16t2 -5 + -31t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -31t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -1.9375t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -1.9375t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -1.9375t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -1.9375t + t2 = 0 + 0.3125 -1.9375t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -1.9375t + t2 = 0.3125 The t term is -1.9375t. Take half its coefficient (-0.96875). Square it (0.9384765625) and add it to both sides. Add '0.9384765625' to each side of the equation. -1.9375t + 0.9384765625 + t2 = 0.3125 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = 0.3125 + 0.9384765625 Combine like terms: 0.3125 + 0.9384765625 = 1.2509765625 0.9384765625 + -1.9375t + t2 = 1.2509765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 1.2509765625 Calculate the square root of the right side: 1.118470636 Break this problem into two subproblems by setting (t + -0.96875) equal to 1.118470636 and -1.118470636.Subproblem 1
t + -0.96875 = 1.118470636 Simplifying t + -0.96875 = 1.118470636 Reorder the terms: -0.96875 + t = 1.118470636 Solving -0.96875 + t = 1.118470636 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = 1.118470636 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 1.118470636 + 0.96875 t = 1.118470636 + 0.96875 Combine like terms: 1.118470636 + 0.96875 = 2.087220636 t = 2.087220636 Simplifying t = 2.087220636Subproblem 2
t + -0.96875 = -1.118470636 Simplifying t + -0.96875 = -1.118470636 Reorder the terms: -0.96875 + t = -1.118470636 Solving -0.96875 + t = -1.118470636 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = -1.118470636 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -1.118470636 + 0.96875 t = -1.118470636 + 0.96875 Combine like terms: -1.118470636 + 0.96875 = -0.149720636 t = -0.149720636 Simplifying t = -0.149720636Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.087220636, -0.149720636}
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