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Simplifying t = -16t2 + 8t + 80 Reorder the terms: t = 80 + 8t + -16t2 Solving t = 80 + 8t + -16t2 Solving for variable 't'. Reorder the terms: -80 + t + -8t + 16t2 = 80 + 8t + -16t2 + -80 + -8t + 16t2 Combine like terms: t + -8t = -7t -80 + -7t + 16t2 = 80 + 8t + -16t2 + -80 + -8t + 16t2 Reorder the terms: -80 + -7t + 16t2 = 80 + -80 + 8t + -8t + -16t2 + 16t2 Combine like terms: 80 + -80 = 0 -80 + -7t + 16t2 = 0 + 8t + -8t + -16t2 + 16t2 -80 + -7t + 16t2 = 8t + -8t + -16t2 + 16t2 Combine like terms: 8t + -8t = 0 -80 + -7t + 16t2 = 0 + -16t2 + 16t2 -80 + -7t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -80 + -7t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -5 + -0.4375t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.4375t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -0.4375t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.4375t + t2 = 0 + 5 -0.4375t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.4375t + t2 = 5 The t term is -0.4375t. Take half its coefficient (-0.21875). Square it (0.0478515625) and add it to both sides. Add '0.0478515625' to each side of the equation. -0.4375t + 0.0478515625 + t2 = 5 + 0.0478515625 Reorder the terms: 0.0478515625 + -0.4375t + t2 = 5 + 0.0478515625 Combine like terms: 5 + 0.0478515625 = 5.0478515625 0.0478515625 + -0.4375t + t2 = 5.0478515625 Factor a perfect square on the left side: (t + -0.21875)(t + -0.21875) = 5.0478515625 Calculate the square root of the right side: 2.246742434 Break this problem into two subproblems by setting (t + -0.21875) equal to 2.246742434 and -2.246742434.Subproblem 1
t + -0.21875 = 2.246742434 Simplifying t + -0.21875 = 2.246742434 Reorder the terms: -0.21875 + t = 2.246742434 Solving -0.21875 + t = 2.246742434 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.21875' to each side of the equation. -0.21875 + 0.21875 + t = 2.246742434 + 0.21875 Combine like terms: -0.21875 + 0.21875 = 0.00000 0.00000 + t = 2.246742434 + 0.21875 t = 2.246742434 + 0.21875 Combine like terms: 2.246742434 + 0.21875 = 2.465492434 t = 2.465492434 Simplifying t = 2.465492434Subproblem 2
t + -0.21875 = -2.246742434 Simplifying t + -0.21875 = -2.246742434 Reorder the terms: -0.21875 + t = -2.246742434 Solving -0.21875 + t = -2.246742434 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.21875' to each side of the equation. -0.21875 + 0.21875 + t = -2.246742434 + 0.21875 Combine like terms: -0.21875 + 0.21875 = 0.00000 0.00000 + t = -2.246742434 + 0.21875 t = -2.246742434 + 0.21875 Combine like terms: -2.246742434 + 0.21875 = -2.027992434 t = -2.027992434 Simplifying t = -2.027992434Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.465492434, -2.027992434}
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