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Simplifying t = 20 + -16t2 + -32t Reorder the terms: t = 20 + -32t + -16t2 Solving t = 20 + -32t + -16t2 Solving for variable 't'. Reorder the terms: -20 + t + 32t + 16t2 = 20 + -32t + -16t2 + -20 + 32t + 16t2 Combine like terms: t + 32t = 33t -20 + 33t + 16t2 = 20 + -32t + -16t2 + -20 + 32t + 16t2 Reorder the terms: -20 + 33t + 16t2 = 20 + -20 + -32t + 32t + -16t2 + 16t2 Combine like terms: 20 + -20 = 0 -20 + 33t + 16t2 = 0 + -32t + 32t + -16t2 + 16t2 -20 + 33t + 16t2 = -32t + 32t + -16t2 + 16t2 Combine like terms: -32t + 32t = 0 -20 + 33t + 16t2 = 0 + -16t2 + 16t2 -20 + 33t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -20 + 33t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.25 + 2.0625t + t2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 2.0625t + 1.25 + t2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 2.0625t + t2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 2.0625t + t2 = 0 + 1.25 2.0625t + t2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 2.0625t + t2 = 1.25 The t term is 2.0625t. Take half its coefficient (1.03125). Square it (1.063476563) and add it to both sides. Add '1.063476563' to each side of the equation. 2.0625t + 1.063476563 + t2 = 1.25 + 1.063476563 Reorder the terms: 1.063476563 + 2.0625t + t2 = 1.25 + 1.063476563 Combine like terms: 1.25 + 1.063476563 = 2.313476563 1.063476563 + 2.0625t + t2 = 2.313476563 Factor a perfect square on the left side: (t + 1.03125)(t + 1.03125) = 2.313476563 Calculate the square root of the right side: 1.521011691 Break this problem into two subproblems by setting (t + 1.03125) equal to 1.521011691 and -1.521011691.Subproblem 1
t + 1.03125 = 1.521011691 Simplifying t + 1.03125 = 1.521011691 Reorder the terms: 1.03125 + t = 1.521011691 Solving 1.03125 + t = 1.521011691 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.03125' to each side of the equation. 1.03125 + -1.03125 + t = 1.521011691 + -1.03125 Combine like terms: 1.03125 + -1.03125 = 0.00000 0.00000 + t = 1.521011691 + -1.03125 t = 1.521011691 + -1.03125 Combine like terms: 1.521011691 + -1.03125 = 0.489761691 t = 0.489761691 Simplifying t = 0.489761691Subproblem 2
t + 1.03125 = -1.521011691 Simplifying t + 1.03125 = -1.521011691 Reorder the terms: 1.03125 + t = -1.521011691 Solving 1.03125 + t = -1.521011691 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.03125' to each side of the equation. 1.03125 + -1.03125 + t = -1.521011691 + -1.03125 Combine like terms: 1.03125 + -1.03125 = 0.00000 0.00000 + t = -1.521011691 + -1.03125 t = -1.521011691 + -1.03125 Combine like terms: -1.521011691 + -1.03125 = -2.552261691 t = -2.552261691 Simplifying t = -2.552261691Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.489761691, -2.552261691}
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