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Simplifying 3 = -16t2 + 30t + 12 Reorder the terms: 3 = 12 + 30t + -16t2 Solving 3 = 12 + 30t + -16t2 Solving for variable 't'. Combine like terms: 3 + -12 = -9 -9 + -30t + 16t2 = 12 + 30t + -16t2 + -12 + -30t + 16t2 Reorder the terms: -9 + -30t + 16t2 = 12 + -12 + 30t + -30t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -9 + -30t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 -9 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 -9 + -30t + 16t2 = 0 + -16t2 + 16t2 -9 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -9 + -30t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.5625 + -1.875t + t2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + -1.875t + 0.5625 + t2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + -1.875t + t2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + -1.875t + t2 = 0 + 0.5625 -1.875t + t2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 -1.875t + t2 = 0.5625 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 = 0.5625 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = 0.5625 + 0.87890625 Combine like terms: 0.5625 + 0.87890625 = 1.44140625 0.87890625 + -1.875t + t2 = 1.44140625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 1.44140625 Calculate the square root of the right side: 1.200585795 Break this problem into two subproblems by setting (t + -0.9375) equal to 1.200585795 and -1.200585795.Subproblem 1
t + -0.9375 = 1.200585795 Simplifying t + -0.9375 = 1.200585795 Reorder the terms: -0.9375 + t = 1.200585795 Solving -0.9375 + t = 1.200585795 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 = 1.200585795 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 1.200585795 + 0.9375 t = 1.200585795 + 0.9375 Combine like terms: 1.200585795 + 0.9375 = 2.138085795 t = 2.138085795 Simplifying t = 2.138085795Subproblem 2
t + -0.9375 = -1.200585795 Simplifying t + -0.9375 = -1.200585795 Reorder the terms: -0.9375 + t = -1.200585795 Solving -0.9375 + t = -1.200585795 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 = -1.200585795 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -1.200585795 + 0.9375 t = -1.200585795 + 0.9375 Combine like terms: -1.200585795 + 0.9375 = -0.263085795 t = -0.263085795 Simplifying t = -0.263085795Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.138085795, -0.263085795}
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