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Simplifying 128 = -16t2 + 128t + 5 Reorder the terms: 128 = 5 + 128t + -16t2 Solving 128 = 5 + 128t + -16t2 Solving for variable 't'. Combine like terms: 128 + -5 = 123 123 + -128t + 16t2 = 5 + 128t + -16t2 + -5 + -128t + 16t2 Reorder the terms: 123 + -128t + 16t2 = 5 + -5 + 128t + -128t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 123 + -128t + 16t2 = 0 + 128t + -128t + -16t2 + 16t2 123 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 123 + -128t + 16t2 = 0 + -16t2 + 16t2 123 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 123 + -128t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 7.6875 + -8t + t2 = 0 Move the constant term to the right: Add '-7.6875' to each side of the equation. 7.6875 + -8t + -7.6875 + t2 = 0 + -7.6875 Reorder the terms: 7.6875 + -7.6875 + -8t + t2 = 0 + -7.6875 Combine like terms: 7.6875 + -7.6875 = 0.0000 0.0000 + -8t + t2 = 0 + -7.6875 -8t + t2 = 0 + -7.6875 Combine like terms: 0 + -7.6875 = -7.6875 -8t + t2 = -7.6875 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = -7.6875 + 16 Reorder the terms: 16 + -8t + t2 = -7.6875 + 16 Combine like terms: -7.6875 + 16 = 8.3125 16 + -8t + t2 = 8.3125 Factor a perfect square on the left side: (t + -4)(t + -4) = 8.3125 Calculate the square root of the right side: 2.883140649 Break this problem into two subproblems by setting (t + -4) equal to 2.883140649 and -2.883140649.Subproblem 1
t + -4 = 2.883140649 Simplifying t + -4 = 2.883140649 Reorder the terms: -4 + t = 2.883140649 Solving -4 + t = 2.883140649 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 2.883140649 + 4 Combine like terms: -4 + 4 = 0 0 + t = 2.883140649 + 4 t = 2.883140649 + 4 Combine like terms: 2.883140649 + 4 = 6.883140649 t = 6.883140649 Simplifying t = 6.883140649Subproblem 2
t + -4 = -2.883140649 Simplifying t + -4 = -2.883140649 Reorder the terms: -4 + t = -2.883140649 Solving -4 + t = -2.883140649 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -2.883140649 + 4 Combine like terms: -4 + 4 = 0 0 + t = -2.883140649 + 4 t = -2.883140649 + 4 Combine like terms: -2.883140649 + 4 = 1.116859351 t = 1.116859351 Simplifying t = 1.116859351Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.883140649, 1.116859351}
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