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Simplifying t = -16t2 + 104t + 56 Reorder the terms: t = 56 + 104t + -16t2 Solving t = 56 + 104t + -16t2 Solving for variable 't'. Reorder the terms: -56 + t + -104t + 16t2 = 56 + 104t + -16t2 + -56 + -104t + 16t2 Combine like terms: t + -104t = -103t -56 + -103t + 16t2 = 56 + 104t + -16t2 + -56 + -104t + 16t2 Reorder the terms: -56 + -103t + 16t2 = 56 + -56 + 104t + -104t + -16t2 + 16t2 Combine like terms: 56 + -56 = 0 -56 + -103t + 16t2 = 0 + 104t + -104t + -16t2 + 16t2 -56 + -103t + 16t2 = 104t + -104t + -16t2 + 16t2 Combine like terms: 104t + -104t = 0 -56 + -103t + 16t2 = 0 + -16t2 + 16t2 -56 + -103t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -56 + -103t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -3.5 + -6.4375t + t2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -6.4375t + 3.5 + t2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -6.4375t + t2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -6.4375t + t2 = 0 + 3.5 -6.4375t + t2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -6.4375t + t2 = 3.5 The t term is -6.4375t. Take half its coefficient (-3.21875). Square it (10.36035156) and add it to both sides. Add '10.36035156' to each side of the equation. -6.4375t + 10.36035156 + t2 = 3.5 + 10.36035156 Reorder the terms: 10.36035156 + -6.4375t + t2 = 3.5 + 10.36035156 Combine like terms: 3.5 + 10.36035156 = 13.86035156 10.36035156 + -6.4375t + t2 = 13.86035156 Factor a perfect square on the left side: (t + -3.21875)(t + -3.21875) = 13.86035156 Calculate the square root of the right side: 3.722949309 Break this problem into two subproblems by setting (t + -3.21875) equal to 3.722949309 and -3.722949309.Subproblem 1
t + -3.21875 = 3.722949309 Simplifying t + -3.21875 = 3.722949309 Reorder the terms: -3.21875 + t = 3.722949309 Solving -3.21875 + t = 3.722949309 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.21875' to each side of the equation. -3.21875 + 3.21875 + t = 3.722949309 + 3.21875 Combine like terms: -3.21875 + 3.21875 = 0.00000 0.00000 + t = 3.722949309 + 3.21875 t = 3.722949309 + 3.21875 Combine like terms: 3.722949309 + 3.21875 = 6.941699309 t = 6.941699309 Simplifying t = 6.941699309Subproblem 2
t + -3.21875 = -3.722949309 Simplifying t + -3.21875 = -3.722949309 Reorder the terms: -3.21875 + t = -3.722949309 Solving -3.21875 + t = -3.722949309 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.21875' to each side of the equation. -3.21875 + 3.21875 + t = -3.722949309 + 3.21875 Combine like terms: -3.21875 + 3.21875 = 0.00000 0.00000 + t = -3.722949309 + 3.21875 t = -3.722949309 + 3.21875 Combine like terms: -3.722949309 + 3.21875 = -0.504199309 t = -0.504199309 Simplifying t = -0.504199309Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.941699309, -0.504199309}
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