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Simplifying 271 = 16t2 + 32t Reorder the terms: 271 = 32t + 16t2 Solving 271 = 32t + 16t2 Solving for variable 't'. Reorder the terms: 271 + -32t + -16t2 = 32t + -32t + 16t2 + -16t2 Combine like terms: 32t + -32t = 0 271 + -32t + -16t2 = 0 + 16t2 + -16t2 271 + -32t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 271 + -32t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -16.9375 + 2t + t2 = 0 Move the constant term to the right: Add '16.9375' to each side of the equation. -16.9375 + 2t + 16.9375 + t2 = 0 + 16.9375 Reorder the terms: -16.9375 + 16.9375 + 2t + t2 = 0 + 16.9375 Combine like terms: -16.9375 + 16.9375 = 0.0000 0.0000 + 2t + t2 = 0 + 16.9375 2t + t2 = 0 + 16.9375 Combine like terms: 0 + 16.9375 = 16.9375 2t + t2 = 16.9375 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 16.9375 + 1 Reorder the terms: 1 + 2t + t2 = 16.9375 + 1 Combine like terms: 16.9375 + 1 = 17.9375 1 + 2t + t2 = 17.9375 Factor a perfect square on the left side: (t + 1)(t + 1) = 17.9375 Calculate the square root of the right side: 4.235268587 Break this problem into two subproblems by setting (t + 1) equal to 4.235268587 and -4.235268587.Subproblem 1
t + 1 = 4.235268587 Simplifying t + 1 = 4.235268587 Reorder the terms: 1 + t = 4.235268587 Solving 1 + t = 4.235268587 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 4.235268587 + -1 Combine like terms: 1 + -1 = 0 0 + t = 4.235268587 + -1 t = 4.235268587 + -1 Combine like terms: 4.235268587 + -1 = 3.235268587 t = 3.235268587 Simplifying t = 3.235268587Subproblem 2
t + 1 = -4.235268587 Simplifying t + 1 = -4.235268587 Reorder the terms: 1 + t = -4.235268587 Solving 1 + t = -4.235268587 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -4.235268587 + -1 Combine like terms: 1 + -1 = 0 0 + t = -4.235268587 + -1 t = -4.235268587 + -1 Combine like terms: -4.235268587 + -1 = -5.235268587 t = -5.235268587 Simplifying t = -5.235268587Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.235268587, -5.235268587}
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