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Simplifying 9(t) = -16t2 + 32t + 5 Reorder the terms: 9t = 5 + 32t + -16t2 Solving 9t = 5 + 32t + -16t2 Solving for variable 't'. Reorder the terms: -5 + 9t + -32t + 16t2 = 5 + 32t + -16t2 + -5 + -32t + 16t2 Combine like terms: 9t + -32t = -23t -5 + -23t + 16t2 = 5 + 32t + -16t2 + -5 + -32t + 16t2 Reorder the terms: -5 + -23t + 16t2 = 5 + -5 + 32t + -32t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -23t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -5 + -23t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -5 + -23t + 16t2 = 0 + -16t2 + 16t2 -5 + -23t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -23t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -1.4375t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -1.4375t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -1.4375t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -1.4375t + t2 = 0 + 0.3125 -1.4375t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -1.4375t + t2 = 0.3125 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 = 0.3125 + 0.5166015625 Reorder the terms: 0.5166015625 + -1.4375t + t2 = 0.3125 + 0.5166015625 Combine like terms: 0.3125 + 0.5166015625 = 0.8291015625 0.5166015625 + -1.4375t + t2 = 0.8291015625 Factor a perfect square on the left side: (t + -0.71875)(t + -0.71875) = 0.8291015625 Calculate the square root of the right side: 0.910550143 Break this problem into two subproblems by setting (t + -0.71875) equal to 0.910550143 and -0.910550143.Subproblem 1
t + -0.71875 = 0.910550143 Simplifying t + -0.71875 = 0.910550143 Reorder the terms: -0.71875 + t = 0.910550143 Solving -0.71875 + t = 0.910550143 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 = 0.910550143 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = 0.910550143 + 0.71875 t = 0.910550143 + 0.71875 Combine like terms: 0.910550143 + 0.71875 = 1.629300143 t = 1.629300143 Simplifying t = 1.629300143Subproblem 2
t + -0.71875 = -0.910550143 Simplifying t + -0.71875 = -0.910550143 Reorder the terms: -0.71875 + t = -0.910550143 Solving -0.71875 + t = -0.910550143 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 = -0.910550143 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = -0.910550143 + 0.71875 t = -0.910550143 + 0.71875 Combine like terms: -0.910550143 + 0.71875 = -0.191800143 t = -0.191800143 Simplifying t = -0.191800143Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.629300143, -0.191800143}
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