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Simplifying 16t2 + 112t + 128 = 5 Reorder the terms: 128 + 112t + 16t2 = 5 Solving 128 + 112t + 16t2 = 5 Solving for variable 't'. Reorder the terms: 128 + -5 + 112t + 16t2 = 5 + -5 Combine like terms: 128 + -5 = 123 123 + 112t + 16t2 = 5 + -5 Combine like terms: 5 + -5 = 0 123 + 112t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 7.6875 + 7t + t2 = 0 Move the constant term to the right: Add '-7.6875' to each side of the equation. 7.6875 + 7t + -7.6875 + t2 = 0 + -7.6875 Reorder the terms: 7.6875 + -7.6875 + 7t + t2 = 0 + -7.6875 Combine like terms: 7.6875 + -7.6875 = 0.0000 0.0000 + 7t + t2 = 0 + -7.6875 7t + t2 = 0 + -7.6875 Combine like terms: 0 + -7.6875 = -7.6875 7t + t2 = -7.6875 The t term is 7t. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7t + 12.25 + t2 = -7.6875 + 12.25 Reorder the terms: 12.25 + 7t + t2 = -7.6875 + 12.25 Combine like terms: -7.6875 + 12.25 = 4.5625 12.25 + 7t + t2 = 4.5625 Factor a perfect square on the left side: (t + 3.5)(t + 3.5) = 4.5625 Calculate the square root of the right side: 2.136000936 Break this problem into two subproblems by setting (t + 3.5) equal to 2.136000936 and -2.136000936.Subproblem 1
t + 3.5 = 2.136000936 Simplifying t + 3.5 = 2.136000936 Reorder the terms: 3.5 + t = 2.136000936 Solving 3.5 + t = 2.136000936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + t = 2.136000936 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + t = 2.136000936 + -3.5 t = 2.136000936 + -3.5 Combine like terms: 2.136000936 + -3.5 = -1.363999064 t = -1.363999064 Simplifying t = -1.363999064Subproblem 2
t + 3.5 = -2.136000936 Simplifying t + 3.5 = -2.136000936 Reorder the terms: 3.5 + t = -2.136000936 Solving 3.5 + t = -2.136000936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + t = -2.136000936 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + t = -2.136000936 + -3.5 t = -2.136000936 + -3.5 Combine like terms: -2.136000936 + -3.5 = -5.636000936 t = -5.636000936 Simplifying t = -5.636000936Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.363999064, -5.636000936}
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