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Simplifying 5.5 = -16t2 + 30t + 14 Reorder the terms: 5.5 = 14 + 30t + -16t2 Solving 5.5 = 14 + 30t + -16t2 Solving for variable 't'. Combine like terms: 5.5 + -14 = -8.5 -8.5 + -30t + 16t2 = 14 + 30t + -16t2 + -14 + -30t + 16t2 Reorder the terms: -8.5 + -30t + 16t2 = 14 + -14 + 30t + -30t + -16t2 + 16t2 Combine like terms: 14 + -14 = 0 -8.5 + -30t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 -8.5 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 -8.5 + -30t + 16t2 = 0 + -16t2 + 16t2 -8.5 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -8.5 + -30t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.53125 + -1.875t + t2 = 0 Move the constant term to the right: Add '0.53125' to each side of the equation. -0.53125 + -1.875t + 0.53125 + t2 = 0 + 0.53125 Reorder the terms: -0.53125 + 0.53125 + -1.875t + t2 = 0 + 0.53125 Combine like terms: -0.53125 + 0.53125 = 0.00000 0.00000 + -1.875t + t2 = 0 + 0.53125 -1.875t + t2 = 0 + 0.53125 Combine like terms: 0 + 0.53125 = 0.53125 -1.875t + t2 = 0.53125 The t term is -1.875t. Take half its coefficient (-0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. -1.875t + 0.87890625 + t2 = 0.53125 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = 0.53125 + 0.87890625 Combine like terms: 0.53125 + 0.87890625 = 1.41015625 0.87890625 + -1.875t + t2 = 1.41015625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 1.41015625 Calculate the square root of the right side: 1.1875 Break this problem into two subproblems by setting (t + -0.9375) equal to 1.1875 and -1.1875.Subproblem 1
t + -0.9375 = 1.1875 Simplifying t + -0.9375 = 1.1875 Reorder the terms: -0.9375 + t = 1.1875 Solving -0.9375 + t = 1.1875 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = 1.1875 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 1.1875 + 0.9375 t = 1.1875 + 0.9375 Combine like terms: 1.1875 + 0.9375 = 2.125 t = 2.125 Simplifying t = 2.125Subproblem 2
t + -0.9375 = -1.1875 Simplifying t + -0.9375 = -1.1875 Reorder the terms: -0.9375 + t = -1.1875 Solving -0.9375 + t = -1.1875 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = -1.1875 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -1.1875 + 0.9375 t = -1.1875 + 0.9375 Combine like terms: -1.1875 + 0.9375 = -0.25 t = -0.25 Simplifying t = -0.25Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.125, -0.25}
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