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Simplifying 20 = 6 + 31t + -16t2 Solving 20 = 6 + 31t + -16t2 Solving for variable 't'. Combine like terms: 20 + -6 = 14 14 + -31t + 16t2 = 6 + 31t + -16t2 + -6 + -31t + 16t2 Reorder the terms: 14 + -31t + 16t2 = 6 + -6 + 31t + -31t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 14 + -31t + 16t2 = 0 + 31t + -31t + -16t2 + 16t2 14 + -31t + 16t2 = 31t + -31t + -16t2 + 16t2 Combine like terms: 31t + -31t = 0 14 + -31t + 16t2 = 0 + -16t2 + 16t2 14 + -31t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 14 + -31t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.875 + -1.9375t + t2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + -1.9375t + -0.875 + t2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + -1.9375t + t2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + -1.9375t + t2 = 0 + -0.875 -1.9375t + t2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 -1.9375t + t2 = -0.875 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 = -0.875 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = -0.875 + 0.9384765625 Combine like terms: -0.875 + 0.9384765625 = 0.0634765625 0.9384765625 + -1.9375t + t2 = 0.0634765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 0.0634765625 Calculate the square root of the right side: 0.251945555 Break this problem into two subproblems by setting (t + -0.96875) equal to 0.251945555 and -0.251945555.Subproblem 1
t + -0.96875 = 0.251945555 Simplifying t + -0.96875 = 0.251945555 Reorder the terms: -0.96875 + t = 0.251945555 Solving -0.96875 + t = 0.251945555 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 = 0.251945555 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 0.251945555 + 0.96875 t = 0.251945555 + 0.96875 Combine like terms: 0.251945555 + 0.96875 = 1.220695555 t = 1.220695555 Simplifying t = 1.220695555Subproblem 2
t + -0.96875 = -0.251945555 Simplifying t + -0.96875 = -0.251945555 Reorder the terms: -0.96875 + t = -0.251945555 Solving -0.96875 + t = -0.251945555 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 = -0.251945555 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -0.251945555 + 0.96875 t = -0.251945555 + 0.96875 Combine like terms: -0.251945555 + 0.96875 = 0.716804445 t = 0.716804445 Simplifying t = 0.716804445Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.220695555, 0.716804445}
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