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Simplifying 19 = -16t2 + 29t + 7 Reorder the terms: 19 = 7 + 29t + -16t2 Solving 19 = 7 + 29t + -16t2 Solving for variable 't'. Combine like terms: 19 + -7 = 12 12 + -29t + 16t2 = 7 + 29t + -16t2 + -7 + -29t + 16t2 Reorder the terms: 12 + -29t + 16t2 = 7 + -7 + 29t + -29t + -16t2 + 16t2 Combine like terms: 7 + -7 = 0 12 + -29t + 16t2 = 0 + 29t + -29t + -16t2 + 16t2 12 + -29t + 16t2 = 29t + -29t + -16t2 + 16t2 Combine like terms: 29t + -29t = 0 12 + -29t + 16t2 = 0 + -16t2 + 16t2 12 + -29t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 12 + -29t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.75 + -1.8125t + t2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -1.8125t + -0.75 + t2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -1.8125t + t2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -1.8125t + t2 = 0 + -0.75 -1.8125t + t2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -1.8125t + t2 = -0.75 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 = -0.75 + 0.8212890625 Reorder the terms: 0.8212890625 + -1.8125t + t2 = -0.75 + 0.8212890625 Combine like terms: -0.75 + 0.8212890625 = 0.0712890625 0.8212890625 + -1.8125t + t2 = 0.0712890625 Factor a perfect square on the left side: (t + -0.90625)(t + -0.90625) = 0.0712890625 Calculate the square root of the right side: 0.267000117 Break this problem into two subproblems by setting (t + -0.90625) equal to 0.267000117 and -0.267000117.Subproblem 1
t + -0.90625 = 0.267000117 Simplifying t + -0.90625 = 0.267000117 Reorder the terms: -0.90625 + t = 0.267000117 Solving -0.90625 + t = 0.267000117 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 = 0.267000117 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = 0.267000117 + 0.90625 t = 0.267000117 + 0.90625 Combine like terms: 0.267000117 + 0.90625 = 1.173250117 t = 1.173250117 Simplifying t = 1.173250117Subproblem 2
t + -0.90625 = -0.267000117 Simplifying t + -0.90625 = -0.267000117 Reorder the terms: -0.90625 + t = -0.267000117 Solving -0.90625 + t = -0.267000117 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 = -0.267000117 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = -0.267000117 + 0.90625 t = -0.267000117 + 0.90625 Combine like terms: -0.267000117 + 0.90625 = 0.639249883 t = 0.639249883 Simplifying t = 0.639249883Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.173250117, 0.639249883}
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