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Simplifying 0 = 16t2 + -31t + -6 Reorder the terms: 0 = -6 + -31t + 16t2 Solving 0 = -6 + -31t + 16t2 Solving for variable 't'. Combine like terms: 0 + 6 = 6 6 + 31t + -16t2 = -6 + -31t + 16t2 + 6 + 31t + -16t2 Reorder the terms: 6 + 31t + -16t2 = -6 + 6 + -31t + 31t + 16t2 + -16t2 Combine like terms: -6 + 6 = 0 6 + 31t + -16t2 = 0 + -31t + 31t + 16t2 + -16t2 6 + 31t + -16t2 = -31t + 31t + 16t2 + -16t2 Combine like terms: -31t + 31t = 0 6 + 31t + -16t2 = 0 + 16t2 + -16t2 6 + 31t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 6 + 31t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.375 + -1.9375t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -1.9375t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -1.9375t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -1.9375t + t2 = 0 + 0.375 -1.9375t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -1.9375t + t2 = 0.375 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 = 0.375 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = 0.375 + 0.9384765625 Combine like terms: 0.375 + 0.9384765625 = 1.3134765625 0.9384765625 + -1.9375t + t2 = 1.3134765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 1.3134765625 Calculate the square root of the right side: 1.146070051 Break this problem into two subproblems by setting (t + -0.96875) equal to 1.146070051 and -1.146070051.Subproblem 1
t + -0.96875 = 1.146070051 Simplifying t + -0.96875 = 1.146070051 Reorder the terms: -0.96875 + t = 1.146070051 Solving -0.96875 + t = 1.146070051 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 = 1.146070051 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 1.146070051 + 0.96875 t = 1.146070051 + 0.96875 Combine like terms: 1.146070051 + 0.96875 = 2.114820051 t = 2.114820051 Simplifying t = 2.114820051Subproblem 2
t + -0.96875 = -1.146070051 Simplifying t + -0.96875 = -1.146070051 Reorder the terms: -0.96875 + t = -1.146070051 Solving -0.96875 + t = -1.146070051 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 = -1.146070051 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -1.146070051 + 0.96875 t = -1.146070051 + 0.96875 Combine like terms: -1.146070051 + 0.96875 = -0.177320051 t = -0.177320051 Simplifying t = -0.177320051Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.114820051, -0.177320051}
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