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Simplifying 16t2 + 96t + 120 = 0 Reorder the terms: 120 + 96t + 16t2 = 0 Solving 120 + 96t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '8'. 8(15 + 12t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(15 + 12t + 2t2)' equal to zero and attempt to solve: Simplifying 15 + 12t + 2t2 = 0 Solving 15 + 12t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 7.5 + 6t + t2 = 0 Move the constant term to the right: Add '-7.5' to each side of the equation. 7.5 + 6t + -7.5 + t2 = 0 + -7.5 Reorder the terms: 7.5 + -7.5 + 6t + t2 = 0 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + 6t + t2 = 0 + -7.5 6t + t2 = 0 + -7.5 Combine like terms: 0 + -7.5 = -7.5 6t + t2 = -7.5 The t term is 6t. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6t + 9 + t2 = -7.5 + 9 Reorder the terms: 9 + 6t + t2 = -7.5 + 9 Combine like terms: -7.5 + 9 = 1.5 9 + 6t + t2 = 1.5 Factor a perfect square on the left side: (t + 3)(t + 3) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (t + 3) equal to 1.224744871 and -1.224744871.Subproblem 1
t + 3 = 1.224744871 Simplifying t + 3 = 1.224744871 Reorder the terms: 3 + t = 1.224744871 Solving 3 + t = 1.224744871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = 1.224744871 + -3 Combine like terms: 3 + -3 = 0 0 + t = 1.224744871 + -3 t = 1.224744871 + -3 Combine like terms: 1.224744871 + -3 = -1.775255129 t = -1.775255129 Simplifying t = -1.775255129Subproblem 2
t + 3 = -1.224744871 Simplifying t + 3 = -1.224744871 Reorder the terms: 3 + t = -1.224744871 Solving 3 + t = -1.224744871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = -1.224744871 + -3 Combine like terms: 3 + -3 = 0 0 + t = -1.224744871 + -3 t = -1.224744871 + -3 Combine like terms: -1.224744871 + -3 = -4.224744871 t = -4.224744871 Simplifying t = -4.224744871Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.775255129, -4.224744871}Solution
t = {-1.775255129, -4.224744871}
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