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Simplifying 0 = -16t2 + 23t + 37 Reorder the terms: 0 = 37 + 23t + -16t2 Solving 0 = 37 + 23t + -16t2 Solving for variable 't'. Combine like terms: 0 + -37 = -37 -37 + -23t + 16t2 = 37 + 23t + -16t2 + -37 + -23t + 16t2 Reorder the terms: -37 + -23t + 16t2 = 37 + -37 + 23t + -23t + -16t2 + 16t2 Combine like terms: 37 + -37 = 0 -37 + -23t + 16t2 = 0 + 23t + -23t + -16t2 + 16t2 -37 + -23t + 16t2 = 23t + -23t + -16t2 + 16t2 Combine like terms: 23t + -23t = 0 -37 + -23t + 16t2 = 0 + -16t2 + 16t2 -37 + -23t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -37 + -23t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.3125 + -1.4375t + t2 = 0 Move the constant term to the right: Add '2.3125' to each side of the equation. -2.3125 + -1.4375t + 2.3125 + t2 = 0 + 2.3125 Reorder the terms: -2.3125 + 2.3125 + -1.4375t + t2 = 0 + 2.3125 Combine like terms: -2.3125 + 2.3125 = 0.0000 0.0000 + -1.4375t + t2 = 0 + 2.3125 -1.4375t + t2 = 0 + 2.3125 Combine like terms: 0 + 2.3125 = 2.3125 -1.4375t + t2 = 2.3125 The t term is -1.4375t. Take half its coefficient (-0.71875). Square it (0.5166015625) and add it to both sides. Add '0.5166015625' to each side of the equation. -1.4375t + 0.5166015625 + t2 = 2.3125 + 0.5166015625 Reorder the terms: 0.5166015625 + -1.4375t + t2 = 2.3125 + 0.5166015625 Combine like terms: 2.3125 + 0.5166015625 = 2.8291015625 0.5166015625 + -1.4375t + t2 = 2.8291015625 Factor a perfect square on the left side: (t + -0.71875)(t + -0.71875) = 2.8291015625 Calculate the square root of the right side: 1.68199333 Break this problem into two subproblems by setting (t + -0.71875) equal to 1.68199333 and -1.68199333.Subproblem 1
t + -0.71875 = 1.68199333 Simplifying t + -0.71875 = 1.68199333 Reorder the terms: -0.71875 + t = 1.68199333 Solving -0.71875 + t = 1.68199333 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + t = 1.68199333 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = 1.68199333 + 0.71875 t = 1.68199333 + 0.71875 Combine like terms: 1.68199333 + 0.71875 = 2.40074333 t = 2.40074333 Simplifying t = 2.40074333Subproblem 2
t + -0.71875 = -1.68199333 Simplifying t + -0.71875 = -1.68199333 Reorder the terms: -0.71875 + t = -1.68199333 Solving -0.71875 + t = -1.68199333 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + t = -1.68199333 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = -1.68199333 + 0.71875 t = -1.68199333 + 0.71875 Combine like terms: -1.68199333 + 0.71875 = -0.96324333 t = -0.96324333 Simplifying t = -0.96324333Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.40074333, -0.96324333}
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