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Simplifying 16t2 + -31t + 7 = 0 Reorder the terms: 7 + -31t + 16t2 = 0 Solving 7 + -31t + 16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.4375 + -1.9375t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -1.9375t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -1.9375t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -1.9375t + t2 = 0 + -0.4375 -1.9375t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -1.9375t + t2 = -0.4375 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 = -0.4375 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = -0.4375 + 0.9384765625 Combine like terms: -0.4375 + 0.9384765625 = 0.5009765625 0.9384765625 + -1.9375t + t2 = 0.5009765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 0.5009765625 Calculate the square root of the right side: 0.707796978 Break this problem into two subproblems by setting (t + -0.96875) equal to 0.707796978 and -0.707796978.Subproblem 1
t + -0.96875 = 0.707796978 Simplifying t + -0.96875 = 0.707796978 Reorder the terms: -0.96875 + t = 0.707796978 Solving -0.96875 + t = 0.707796978 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 = 0.707796978 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 0.707796978 + 0.96875 t = 0.707796978 + 0.96875 Combine like terms: 0.707796978 + 0.96875 = 1.676546978 t = 1.676546978 Simplifying t = 1.676546978Subproblem 2
t + -0.96875 = -0.707796978 Simplifying t + -0.96875 = -0.707796978 Reorder the terms: -0.96875 + t = -0.707796978 Solving -0.96875 + t = -0.707796978 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 = -0.707796978 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -0.707796978 + 0.96875 t = -0.707796978 + 0.96875 Combine like terms: -0.707796978 + 0.96875 = 0.260953022 t = 0.260953022 Simplifying t = 0.260953022Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.676546978, 0.260953022}
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