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Simplifying 0 = 5 + 27t + -16t2 Solving 0 = 5 + 27t + -16t2 Solving for variable 't'. Combine like terms: 0 + -5 = -5 -5 + -27t + 16t2 = 5 + 27t + -16t2 + -5 + -27t + 16t2 Reorder the terms: -5 + -27t + 16t2 = 5 + -5 + 27t + -27t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -27t + 16t2 = 0 + 27t + -27t + -16t2 + 16t2 -5 + -27t + 16t2 = 27t + -27t + -16t2 + 16t2 Combine like terms: 27t + -27t = 0 -5 + -27t + 16t2 = 0 + -16t2 + 16t2 -5 + -27t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -27t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -1.6875t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -1.6875t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -1.6875t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -1.6875t + t2 = 0 + 0.3125 -1.6875t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -1.6875t + t2 = 0.3125 The t term is -1.6875t. Take half its coefficient (-0.84375). Square it (0.7119140625) and add it to both sides. Add '0.7119140625' to each side of the equation. -1.6875t + 0.7119140625 + t2 = 0.3125 + 0.7119140625 Reorder the terms: 0.7119140625 + -1.6875t + t2 = 0.3125 + 0.7119140625 Combine like terms: 0.3125 + 0.7119140625 = 1.0244140625 0.7119140625 + -1.6875t + t2 = 1.0244140625 Factor a perfect square on the left side: (t + -0.84375)(t + -0.84375) = 1.0244140625 Calculate the square root of the right side: 1.012133421 Break this problem into two subproblems by setting (t + -0.84375) equal to 1.012133421 and -1.012133421.Subproblem 1
t + -0.84375 = 1.012133421 Simplifying t + -0.84375 = 1.012133421 Reorder the terms: -0.84375 + t = 1.012133421 Solving -0.84375 + t = 1.012133421 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + t = 1.012133421 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + t = 1.012133421 + 0.84375 t = 1.012133421 + 0.84375 Combine like terms: 1.012133421 + 0.84375 = 1.855883421 t = 1.855883421 Simplifying t = 1.855883421Subproblem 2
t + -0.84375 = -1.012133421 Simplifying t + -0.84375 = -1.012133421 Reorder the terms: -0.84375 + t = -1.012133421 Solving -0.84375 + t = -1.012133421 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + t = -1.012133421 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + t = -1.012133421 + 0.84375 t = -1.012133421 + 0.84375 Combine like terms: -1.012133421 + 0.84375 = -0.168383421 t = -0.168383421 Simplifying t = -0.168383421Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.855883421, -0.168383421}
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