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Simplifying 0 = -16t2 + 30t + 135 Reorder the terms: 0 = 135 + 30t + -16t2 Solving 0 = 135 + 30t + -16t2 Solving for variable 't'. Combine like terms: 0 + -135 = -135 -135 + -30t + 16t2 = 135 + 30t + -16t2 + -135 + -30t + 16t2 Reorder the terms: -135 + -30t + 16t2 = 135 + -135 + 30t + -30t + -16t2 + 16t2 Combine like terms: 135 + -135 = 0 -135 + -30t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 -135 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 -135 + -30t + 16t2 = 0 + -16t2 + 16t2 -135 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -135 + -30t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -8.4375 + -1.875t + t2 = 0 Move the constant term to the right: Add '8.4375' to each side of the equation. -8.4375 + -1.875t + 8.4375 + t2 = 0 + 8.4375 Reorder the terms: -8.4375 + 8.4375 + -1.875t + t2 = 0 + 8.4375 Combine like terms: -8.4375 + 8.4375 = 0.0000 0.0000 + -1.875t + t2 = 0 + 8.4375 -1.875t + t2 = 0 + 8.4375 Combine like terms: 0 + 8.4375 = 8.4375 -1.875t + t2 = 8.4375 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 = 8.4375 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = 8.4375 + 0.87890625 Combine like terms: 8.4375 + 0.87890625 = 9.31640625 0.87890625 + -1.875t + t2 = 9.31640625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 9.31640625 Calculate the square root of the right side: 3.052278862 Break this problem into two subproblems by setting (t + -0.9375) equal to 3.052278862 and -3.052278862.Subproblem 1
t + -0.9375 = 3.052278862 Simplifying t + -0.9375 = 3.052278862 Reorder the terms: -0.9375 + t = 3.052278862 Solving -0.9375 + t = 3.052278862 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 = 3.052278862 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 3.052278862 + 0.9375 t = 3.052278862 + 0.9375 Combine like terms: 3.052278862 + 0.9375 = 3.989778862 t = 3.989778862 Simplifying t = 3.989778862Subproblem 2
t + -0.9375 = -3.052278862 Simplifying t + -0.9375 = -3.052278862 Reorder the terms: -0.9375 + t = -3.052278862 Solving -0.9375 + t = -3.052278862 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 = -3.052278862 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -3.052278862 + 0.9375 t = -3.052278862 + 0.9375 Combine like terms: -3.052278862 + 0.9375 = -2.114778862 t = -2.114778862 Simplifying t = -2.114778862Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.989778862, -2.114778862}
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