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Simplifying -14 = 3 + 31t + -16t2 Solving -14 = 3 + 31t + -16t2 Solving for variable 't'. Combine like terms: -14 + -3 = -17 -17 + -31t + 16t2 = 3 + 31t + -16t2 + -3 + -31t + 16t2 Reorder the terms: -17 + -31t + 16t2 = 3 + -3 + 31t + -31t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 -17 + -31t + 16t2 = 0 + 31t + -31t + -16t2 + 16t2 -17 + -31t + 16t2 = 31t + -31t + -16t2 + 16t2 Combine like terms: 31t + -31t = 0 -17 + -31t + 16t2 = 0 + -16t2 + 16t2 -17 + -31t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -17 + -31t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.0625 + -1.9375t + t2 = 0 Move the constant term to the right: Add '1.0625' to each side of the equation. -1.0625 + -1.9375t + 1.0625 + t2 = 0 + 1.0625 Reorder the terms: -1.0625 + 1.0625 + -1.9375t + t2 = 0 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + -1.9375t + t2 = 0 + 1.0625 -1.9375t + t2 = 0 + 1.0625 Combine like terms: 0 + 1.0625 = 1.0625 -1.9375t + t2 = 1.0625 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 = 1.0625 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = 1.0625 + 0.9384765625 Combine like terms: 1.0625 + 0.9384765625 = 2.0009765625 0.9384765625 + -1.9375t + t2 = 2.0009765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 2.0009765625 Calculate the square root of the right side: 1.414558787 Break this problem into two subproblems by setting (t + -0.96875) equal to 1.414558787 and -1.414558787.Subproblem 1
t + -0.96875 = 1.414558787 Simplifying t + -0.96875 = 1.414558787 Reorder the terms: -0.96875 + t = 1.414558787 Solving -0.96875 + t = 1.414558787 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 = 1.414558787 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 1.414558787 + 0.96875 t = 1.414558787 + 0.96875 Combine like terms: 1.414558787 + 0.96875 = 2.383308787 t = 2.383308787 Simplifying t = 2.383308787Subproblem 2
t + -0.96875 = -1.414558787 Simplifying t + -0.96875 = -1.414558787 Reorder the terms: -0.96875 + t = -1.414558787 Solving -0.96875 + t = -1.414558787 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 = -1.414558787 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -1.414558787 + 0.96875 t = -1.414558787 + 0.96875 Combine like terms: -1.414558787 + 0.96875 = -0.445808787 t = -0.445808787 Simplifying t = -0.445808787Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.383308787, -0.445808787}
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