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Simplifying 6 = -16t2 + 31t + 4 Reorder the terms: 6 = 4 + 31t + -16t2 Solving 6 = 4 + 31t + -16t2 Solving for variable 't'. Combine like terms: 6 + -4 = 2 2 + -31t + 16t2 = 4 + 31t + -16t2 + -4 + -31t + 16t2 Reorder the terms: 2 + -31t + 16t2 = 4 + -4 + 31t + -31t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 2 + -31t + 16t2 = 0 + 31t + -31t + -16t2 + 16t2 2 + -31t + 16t2 = 31t + -31t + -16t2 + 16t2 Combine like terms: 31t + -31t = 0 2 + -31t + 16t2 = 0 + -16t2 + 16t2 2 + -31t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 2 + -31t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.125 + -1.9375t + t2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -1.9375t + -0.125 + t2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -1.9375t + t2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -1.9375t + t2 = 0 + -0.125 -1.9375t + t2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -1.9375t + t2 = -0.125 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 = -0.125 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = -0.125 + 0.9384765625 Combine like terms: -0.125 + 0.9384765625 = 0.8134765625 0.9384765625 + -1.9375t + t2 = 0.8134765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 0.8134765625 Calculate the square root of the right side: 0.901929356 Break this problem into two subproblems by setting (t + -0.96875) equal to 0.901929356 and -0.901929356.Subproblem 1
t + -0.96875 = 0.901929356 Simplifying t + -0.96875 = 0.901929356 Reorder the terms: -0.96875 + t = 0.901929356 Solving -0.96875 + t = 0.901929356 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 = 0.901929356 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 0.901929356 + 0.96875 t = 0.901929356 + 0.96875 Combine like terms: 0.901929356 + 0.96875 = 1.870679356 t = 1.870679356 Simplifying t = 1.870679356Subproblem 2
t + -0.96875 = -0.901929356 Simplifying t + -0.96875 = -0.901929356 Reorder the terms: -0.96875 + t = -0.901929356 Solving -0.96875 + t = -0.901929356 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 = -0.901929356 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -0.901929356 + 0.96875 t = -0.901929356 + 0.96875 Combine like terms: -0.901929356 + 0.96875 = 0.066820644 t = 0.066820644 Simplifying t = 0.066820644Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.870679356, 0.066820644}
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