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Simplifying -16t2 + 27t + 2.5 = 3.5 Reorder the terms: 2.5 + 27t + -16t2 = 3.5 Solving 2.5 + 27t + -16t2 = 3.5 Solving for variable 't'. Reorder the terms: 2.5 + -3.5 + 27t + -16t2 = 3.5 + -3.5 Combine like terms: 2.5 + -3.5 = -1 -1 + 27t + -16t2 = 3.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 -1 + 27t + -16t2 = 0.0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.0625 + -1.6875t + t2 = 0.0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + -1.6875t + -0.0625 + t2 = 0.0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + -1.6875t + t2 = 0.0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + -1.6875t + t2 = 0.0 + -0.0625 -1.6875t + t2 = 0.0 + -0.0625 Combine like terms: 0.0 + -0.0625 = -0.0625 -1.6875t + t2 = -0.0625 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 = -0.0625 + 0.7119140625 Reorder the terms: 0.7119140625 + -1.6875t + t2 = -0.0625 + 0.7119140625 Combine like terms: -0.0625 + 0.7119140625 = 0.6494140625 0.7119140625 + -1.6875t + t2 = 0.6494140625 Factor a perfect square on the left side: (t + -0.84375)(t + -0.84375) = 0.6494140625 Calculate the square root of the right side: 0.80586231 Break this problem into two subproblems by setting (t + -0.84375) equal to 0.80586231 and -0.80586231.Subproblem 1
t + -0.84375 = 0.80586231 Simplifying t + -0.84375 = 0.80586231 Reorder the terms: -0.84375 + t = 0.80586231 Solving -0.84375 + t = 0.80586231 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 = 0.80586231 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + t = 0.80586231 + 0.84375 t = 0.80586231 + 0.84375 Combine like terms: 0.80586231 + 0.84375 = 1.64961231 t = 1.64961231 Simplifying t = 1.64961231Subproblem 2
t + -0.84375 = -0.80586231 Simplifying t + -0.84375 = -0.80586231 Reorder the terms: -0.84375 + t = -0.80586231 Solving -0.84375 + t = -0.80586231 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 = -0.80586231 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + t = -0.80586231 + 0.84375 t = -0.80586231 + 0.84375 Combine like terms: -0.80586231 + 0.84375 = 0.03788769 t = 0.03788769 Simplifying t = 0.03788769Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.64961231, 0.03788769}
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