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Simplifying 0 = -16t2 + -29t + 180 Reorder the terms: 0 = 180 + -29t + -16t2 Solving 0 = 180 + -29t + -16t2 Solving for variable 't'. Combine like terms: 0 + -180 = -180 -180 + 29t + 16t2 = 180 + -29t + -16t2 + -180 + 29t + 16t2 Reorder the terms: -180 + 29t + 16t2 = 180 + -180 + -29t + 29t + -16t2 + 16t2 Combine like terms: 180 + -180 = 0 -180 + 29t + 16t2 = 0 + -29t + 29t + -16t2 + 16t2 -180 + 29t + 16t2 = -29t + 29t + -16t2 + 16t2 Combine like terms: -29t + 29t = 0 -180 + 29t + 16t2 = 0 + -16t2 + 16t2 -180 + 29t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -180 + 29t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -11.25 + 1.8125t + t2 = 0 Move the constant term to the right: Add '11.25' to each side of the equation. -11.25 + 1.8125t + 11.25 + t2 = 0 + 11.25 Reorder the terms: -11.25 + 11.25 + 1.8125t + t2 = 0 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + 1.8125t + t2 = 0 + 11.25 1.8125t + t2 = 0 + 11.25 Combine like terms: 0 + 11.25 = 11.25 1.8125t + t2 = 11.25 The t term is 1.8125t. Take half its coefficient (0.90625). Square it (0.8212890625) and add it to both sides. Add '0.8212890625' to each side of the equation. 1.8125t + 0.8212890625 + t2 = 11.25 + 0.8212890625 Reorder the terms: 0.8212890625 + 1.8125t + t2 = 11.25 + 0.8212890625 Combine like terms: 11.25 + 0.8212890625 = 12.0712890625 0.8212890625 + 1.8125t + t2 = 12.0712890625 Factor a perfect square on the left side: (t + 0.90625)(t + 0.90625) = 12.0712890625 Calculate the square root of the right side: 3.474376068 Break this problem into two subproblems by setting (t + 0.90625) equal to 3.474376068 and -3.474376068.Subproblem 1
t + 0.90625 = 3.474376068 Simplifying t + 0.90625 = 3.474376068 Reorder the terms: 0.90625 + t = 3.474376068 Solving 0.90625 + t = 3.474376068 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + t = 3.474376068 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + t = 3.474376068 + -0.90625 t = 3.474376068 + -0.90625 Combine like terms: 3.474376068 + -0.90625 = 2.568126068 t = 2.568126068 Simplifying t = 2.568126068Subproblem 2
t + 0.90625 = -3.474376068 Simplifying t + 0.90625 = -3.474376068 Reorder the terms: 0.90625 + t = -3.474376068 Solving 0.90625 + t = -3.474376068 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + t = -3.474376068 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + t = -3.474376068 + -0.90625 t = -3.474376068 + -0.90625 Combine like terms: -3.474376068 + -0.90625 = -4.380626068 t = -4.380626068 Simplifying t = -4.380626068Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.568126068, -4.380626068}
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