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Simplifying 162(t) = 88t + -11t2 + 30 Reorder the terms: 162t = 30 + 88t + -11t2 Solving 162t = 30 + 88t + -11t2 Solving for variable 't'. Reorder the terms: -30 + 162t + -88t + 11t2 = 30 + 88t + -11t2 + -30 + -88t + 11t2 Combine like terms: 162t + -88t = 74t -30 + 74t + 11t2 = 30 + 88t + -11t2 + -30 + -88t + 11t2 Reorder the terms: -30 + 74t + 11t2 = 30 + -30 + 88t + -88t + -11t2 + 11t2 Combine like terms: 30 + -30 = 0 -30 + 74t + 11t2 = 0 + 88t + -88t + -11t2 + 11t2 -30 + 74t + 11t2 = 88t + -88t + -11t2 + 11t2 Combine like terms: 88t + -88t = 0 -30 + 74t + 11t2 = 0 + -11t2 + 11t2 -30 + 74t + 11t2 = -11t2 + 11t2 Combine like terms: -11t2 + 11t2 = 0 -30 + 74t + 11t2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -2.727272727 + 6.727272727t + t2 = 0 Move the constant term to the right: Add '2.727272727' to each side of the equation. -2.727272727 + 6.727272727t + 2.727272727 + t2 = 0 + 2.727272727 Reorder the terms: -2.727272727 + 2.727272727 + 6.727272727t + t2 = 0 + 2.727272727 Combine like terms: -2.727272727 + 2.727272727 = 0.000000000 0.000000000 + 6.727272727t + t2 = 0 + 2.727272727 6.727272727t + t2 = 0 + 2.727272727 Combine like terms: 0 + 2.727272727 = 2.727272727 6.727272727t + t2 = 2.727272727 The t term is 6.727272727t. Take half its coefficient (3.363636364). Square it (11.31404959) and add it to both sides. Add '11.31404959' to each side of the equation. 6.727272727t + 11.31404959 + t2 = 2.727272727 + 11.31404959 Reorder the terms: 11.31404959 + 6.727272727t + t2 = 2.727272727 + 11.31404959 Combine like terms: 2.727272727 + 11.31404959 = 14.041322317 11.31404959 + 6.727272727t + t2 = 14.041322317 Factor a perfect square on the left side: (t + 3.363636364)(t + 3.363636364) = 14.041322317 Calculate the square root of the right side: 3.747175245 Break this problem into two subproblems by setting (t + 3.363636364) equal to 3.747175245 and -3.747175245.Subproblem 1
t + 3.363636364 = 3.747175245 Simplifying t + 3.363636364 = 3.747175245 Reorder the terms: 3.363636364 + t = 3.747175245 Solving 3.363636364 + t = 3.747175245 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.363636364' to each side of the equation. 3.363636364 + -3.363636364 + t = 3.747175245 + -3.363636364 Combine like terms: 3.363636364 + -3.363636364 = 0.000000000 0.000000000 + t = 3.747175245 + -3.363636364 t = 3.747175245 + -3.363636364 Combine like terms: 3.747175245 + -3.363636364 = 0.383538881 t = 0.383538881 Simplifying t = 0.383538881Subproblem 2
t + 3.363636364 = -3.747175245 Simplifying t + 3.363636364 = -3.747175245 Reorder the terms: 3.363636364 + t = -3.747175245 Solving 3.363636364 + t = -3.747175245 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.363636364' to each side of the equation. 3.363636364 + -3.363636364 + t = -3.747175245 + -3.363636364 Combine like terms: 3.363636364 + -3.363636364 = 0.000000000 0.000000000 + t = -3.747175245 + -3.363636364 t = -3.747175245 + -3.363636364 Combine like terms: -3.747175245 + -3.363636364 = -7.110811609 t = -7.110811609 Simplifying t = -7.110811609Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.383538881, -7.110811609}
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