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Simplifying 0 = 16t2 + 31t + 2 Reorder the terms: 0 = 2 + 31t + 16t2 Solving 0 = 2 + 31t + 16t2 Solving for variable 't'. Combine like terms: 0 + -2 = -2 -2 + -31t + -16t2 = 2 + 31t + 16t2 + -2 + -31t + -16t2 Reorder the terms: -2 + -31t + -16t2 = 2 + -2 + 31t + -31t + 16t2 + -16t2 Combine like terms: 2 + -2 = 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 Factor out the Greatest Common Factor (GCF), '-1'. -1(2 + 31t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(2 + 31t + 16t2)' equal to zero and attempt to solve: Simplifying 2 + 31t + 16t2 = 0 Solving 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 = -0.066820644 t = -0.066820644 Simplifying t = -0.066820644Subproblem 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 = -1.870679356 t = -1.870679356 Simplifying t = -1.870679356Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.066820644, -1.870679356}Solution
t = {-0.066820644, -1.870679356}
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