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Simplifying 0 = 16t2 + 116t + 101 Reorder the terms: 0 = 101 + 116t + 16t2 Solving 0 = 101 + 116t + 16t2 Solving for variable 't'. Combine like terms: 0 + -101 = -101 -101 + -116t + -16t2 = 101 + 116t + 16t2 + -101 + -116t + -16t2 Reorder the terms: -101 + -116t + -16t2 = 101 + -101 + 116t + -116t + 16t2 + -16t2 Combine like terms: 101 + -101 = 0 -101 + -116t + -16t2 = 0 + 116t + -116t + 16t2 + -16t2 -101 + -116t + -16t2 = 116t + -116t + 16t2 + -16t2 Combine like terms: 116t + -116t = 0 -101 + -116t + -16t2 = 0 + 16t2 + -16t2 -101 + -116t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -101 + -116t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(101 + 116t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(101 + 116t + 16t2)' equal to zero and attempt to solve: Simplifying 101 + 116t + 16t2 = 0 Solving 101 + 116t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 6.3125 + 7.25t + t2 = 0 Move the constant term to the right: Add '-6.3125' to each side of the equation. 6.3125 + 7.25t + -6.3125 + t2 = 0 + -6.3125 Reorder the terms: 6.3125 + -6.3125 + 7.25t + t2 = 0 + -6.3125 Combine like terms: 6.3125 + -6.3125 = 0.0000 0.0000 + 7.25t + t2 = 0 + -6.3125 7.25t + t2 = 0 + -6.3125 Combine like terms: 0 + -6.3125 = -6.3125 7.25t + t2 = -6.3125 The t term is 7.25t. Take half its coefficient (3.625). Square it (13.140625) and add it to both sides. Add '13.140625' to each side of the equation. 7.25t + 13.140625 + t2 = -6.3125 + 13.140625 Reorder the terms: 13.140625 + 7.25t + t2 = -6.3125 + 13.140625 Combine like terms: -6.3125 + 13.140625 = 6.828125 13.140625 + 7.25t + t2 = 6.828125 Factor a perfect square on the left side: (t + 3.625)(t + 3.625) = 6.828125 Calculate the square root of the right side: 2.61306812 Break this problem into two subproblems by setting (t + 3.625) equal to 2.61306812 and -2.61306812.Subproblem 1
t + 3.625 = 2.61306812 Simplifying t + 3.625 = 2.61306812 Reorder the terms: 3.625 + t = 2.61306812 Solving 3.625 + t = 2.61306812 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.625' to each side of the equation. 3.625 + -3.625 + t = 2.61306812 + -3.625 Combine like terms: 3.625 + -3.625 = 0.000 0.000 + t = 2.61306812 + -3.625 t = 2.61306812 + -3.625 Combine like terms: 2.61306812 + -3.625 = -1.01193188 t = -1.01193188 Simplifying t = -1.01193188Subproblem 2
t + 3.625 = -2.61306812 Simplifying t + 3.625 = -2.61306812 Reorder the terms: 3.625 + t = -2.61306812 Solving 3.625 + t = -2.61306812 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.625' to each side of the equation. 3.625 + -3.625 + t = -2.61306812 + -3.625 Combine like terms: 3.625 + -3.625 = 0.000 0.000 + t = -2.61306812 + -3.625 t = -2.61306812 + -3.625 Combine like terms: -2.61306812 + -3.625 = -6.23806812 t = -6.23806812 Simplifying t = -6.23806812Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.01193188, -6.23806812}Solution
t = {-1.01193188, -6.23806812}
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