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Simplifying 63t = -16t2 + 107t + 63 Reorder the terms: 63t = 63 + 107t + -16t2 Solving 63t = 63 + 107t + -16t2 Solving for variable 't'. Reorder the terms: -63 + 63t + -107t + 16t2 = 63 + 107t + -16t2 + -63 + -107t + 16t2 Combine like terms: 63t + -107t = -44t -63 + -44t + 16t2 = 63 + 107t + -16t2 + -63 + -107t + 16t2 Reorder the terms: -63 + -44t + 16t2 = 63 + -63 + 107t + -107t + -16t2 + 16t2 Combine like terms: 63 + -63 = 0 -63 + -44t + 16t2 = 0 + 107t + -107t + -16t2 + 16t2 -63 + -44t + 16t2 = 107t + -107t + -16t2 + 16t2 Combine like terms: 107t + -107t = 0 -63 + -44t + 16t2 = 0 + -16t2 + 16t2 -63 + -44t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -63 + -44t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -3.9375 + -2.75t + t2 = 0 Move the constant term to the right: Add '3.9375' to each side of the equation. -3.9375 + -2.75t + 3.9375 + t2 = 0 + 3.9375 Reorder the terms: -3.9375 + 3.9375 + -2.75t + t2 = 0 + 3.9375 Combine like terms: -3.9375 + 3.9375 = 0.0000 0.0000 + -2.75t + t2 = 0 + 3.9375 -2.75t + t2 = 0 + 3.9375 Combine like terms: 0 + 3.9375 = 3.9375 -2.75t + t2 = 3.9375 The t term is -2.75t. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75t + 1.890625 + t2 = 3.9375 + 1.890625 Reorder the terms: 1.890625 + -2.75t + t2 = 3.9375 + 1.890625 Combine like terms: 3.9375 + 1.890625 = 5.828125 1.890625 + -2.75t + t2 = 5.828125 Factor a perfect square on the left side: (t + -1.375)(t + -1.375) = 5.828125 Calculate the square root of the right side: 2.414150989 Break this problem into two subproblems by setting (t + -1.375) equal to 2.414150989 and -2.414150989.Subproblem 1
t + -1.375 = 2.414150989 Simplifying t + -1.375 = 2.414150989 Reorder the terms: -1.375 + t = 2.414150989 Solving -1.375 + t = 2.414150989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + t = 2.414150989 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + t = 2.414150989 + 1.375 t = 2.414150989 + 1.375 Combine like terms: 2.414150989 + 1.375 = 3.789150989 t = 3.789150989 Simplifying t = 3.789150989Subproblem 2
t + -1.375 = -2.414150989 Simplifying t + -1.375 = -2.414150989 Reorder the terms: -1.375 + t = -2.414150989 Solving -1.375 + t = -2.414150989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + t = -2.414150989 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + t = -2.414150989 + 1.375 t = -2.414150989 + 1.375 Combine like terms: -2.414150989 + 1.375 = -1.039150989 t = -1.039150989 Simplifying t = -1.039150989Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.789150989, -1.039150989}
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