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Simplifying 40t = -16t2 + 70t + 5 Reorder the terms: 40t = 5 + 70t + -16t2 Solving 40t = 5 + 70t + -16t2 Solving for variable 't'. Reorder the terms: -5 + 40t + -70t + 16t2 = 5 + 70t + -16t2 + -5 + -70t + 16t2 Combine like terms: 40t + -70t = -30t -5 + -30t + 16t2 = 5 + 70t + -16t2 + -5 + -70t + 16t2 Reorder the terms: -5 + -30t + 16t2 = 5 + -5 + 70t + -70t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -30t + 16t2 = 0 + 70t + -70t + -16t2 + 16t2 -5 + -30t + 16t2 = 70t + -70t + -16t2 + 16t2 Combine like terms: 70t + -70t = 0 -5 + -30t + 16t2 = 0 + -16t2 + 16t2 -5 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -30t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -1.875t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -1.875t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -1.875t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -1.875t + t2 = 0 + 0.3125 -1.875t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -1.875t + t2 = 0.3125 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 = 0.3125 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = 0.3125 + 0.87890625 Combine like terms: 0.3125 + 0.87890625 = 1.19140625 0.87890625 + -1.875t + t2 = 1.19140625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 1.19140625 Calculate the square root of the right side: 1.091515575 Break this problem into two subproblems by setting (t + -0.9375) equal to 1.091515575 and -1.091515575.Subproblem 1
t + -0.9375 = 1.091515575 Simplifying t + -0.9375 = 1.091515575 Reorder the terms: -0.9375 + t = 1.091515575 Solving -0.9375 + t = 1.091515575 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 = 1.091515575 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 1.091515575 + 0.9375 t = 1.091515575 + 0.9375 Combine like terms: 1.091515575 + 0.9375 = 2.029015575 t = 2.029015575 Simplifying t = 2.029015575Subproblem 2
t + -0.9375 = -1.091515575 Simplifying t + -0.9375 = -1.091515575 Reorder the terms: -0.9375 + t = -1.091515575 Solving -0.9375 + t = -1.091515575 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 = -1.091515575 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -1.091515575 + 0.9375 t = -1.091515575 + 0.9375 Combine like terms: -1.091515575 + 0.9375 = -0.154015575 t = -0.154015575 Simplifying t = -0.154015575Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.029015575, -0.154015575}
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