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Simplifying 3 = -16t2 + 30t Reorder the terms: 3 = 30t + -16t2 Solving 3 = 30t + -16t2 Solving for variable 't'. Reorder the terms: 3 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 3 + -30t + 16t2 = 0 + -16t2 + 16t2 3 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 3 + -30t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + -1.875t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -1.875t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -1.875t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -1.875t + t2 = 0 + -0.1875 -1.875t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -1.875t + t2 = -0.1875 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 = -0.1875 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = -0.1875 + 0.87890625 Combine like terms: -0.1875 + 0.87890625 = 0.69140625 0.87890625 + -1.875t + t2 = 0.69140625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 0.69140625 Calculate the square root of the right side: 0.831508418 Break this problem into two subproblems by setting (t + -0.9375) equal to 0.831508418 and -0.831508418.Subproblem 1
t + -0.9375 = 0.831508418 Simplifying t + -0.9375 = 0.831508418 Reorder the terms: -0.9375 + t = 0.831508418 Solving -0.9375 + t = 0.831508418 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 = 0.831508418 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 0.831508418 + 0.9375 t = 0.831508418 + 0.9375 Combine like terms: 0.831508418 + 0.9375 = 1.769008418 t = 1.769008418 Simplifying t = 1.769008418Subproblem 2
t + -0.9375 = -0.831508418 Simplifying t + -0.9375 = -0.831508418 Reorder the terms: -0.9375 + t = -0.831508418 Solving -0.9375 + t = -0.831508418 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 = -0.831508418 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -0.831508418 + 0.9375 t = -0.831508418 + 0.9375 Combine like terms: -0.831508418 + 0.9375 = 0.105991582 t = 0.105991582 Simplifying t = 0.105991582Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.769008418, 0.105991582}
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