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Simplifying 8 = -16t2 + 60t + 1 Reorder the terms: 8 = 1 + 60t + -16t2 Solving 8 = 1 + 60t + -16t2 Solving for variable 't'. Combine like terms: 8 + -1 = 7 7 + -60t + 16t2 = 1 + 60t + -16t2 + -1 + -60t + 16t2 Reorder the terms: 7 + -60t + 16t2 = 1 + -1 + 60t + -60t + -16t2 + 16t2 Combine like terms: 1 + -1 = 0 7 + -60t + 16t2 = 0 + 60t + -60t + -16t2 + 16t2 7 + -60t + 16t2 = 60t + -60t + -16t2 + 16t2 Combine like terms: 60t + -60t = 0 7 + -60t + 16t2 = 0 + -16t2 + 16t2 7 + -60t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 7 + -60t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.4375 + -3.75t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -3.75t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -3.75t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -3.75t + t2 = 0 + -0.4375 -3.75t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -3.75t + t2 = -0.4375 The t term is -3.75t. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75t + 3.515625 + t2 = -0.4375 + 3.515625 Reorder the terms: 3.515625 + -3.75t + t2 = -0.4375 + 3.515625 Combine like terms: -0.4375 + 3.515625 = 3.078125 3.515625 + -3.75t + t2 = 3.078125 Factor a perfect square on the left side: (t + -1.875)(t + -1.875) = 3.078125 Calculate the square root of the right side: 1.754458606 Break this problem into two subproblems by setting (t + -1.875) equal to 1.754458606 and -1.754458606.Subproblem 1
t + -1.875 = 1.754458606 Simplifying t + -1.875 = 1.754458606 Reorder the terms: -1.875 + t = 1.754458606 Solving -1.875 + t = 1.754458606 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + t = 1.754458606 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + t = 1.754458606 + 1.875 t = 1.754458606 + 1.875 Combine like terms: 1.754458606 + 1.875 = 3.629458606 t = 3.629458606 Simplifying t = 3.629458606Subproblem 2
t + -1.875 = -1.754458606 Simplifying t + -1.875 = -1.754458606 Reorder the terms: -1.875 + t = -1.754458606 Solving -1.875 + t = -1.754458606 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + t = -1.754458606 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + t = -1.754458606 + 1.875 t = -1.754458606 + 1.875 Combine like terms: -1.754458606 + 1.875 = 0.120541394 t = 0.120541394 Simplifying t = 0.120541394Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.629458606, 0.120541394}
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