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Simplifying (2t)(t + -6) = (18)(t + -6) + 3 Remove parenthesis around (2t) 2t(t + -6) = (18)(t + -6) + 3 Reorder the terms: 2t(-6 + t) = (18)(t + -6) + 3 (-6 * 2t + t * 2t) = (18)(t + -6) + 3 (-12t + 2t2) = (18)(t + -6) + 3 Reorder the terms: -12t + 2t2 = 18(-6 + t) + 3 -12t + 2t2 = (-6 * 18 + t * 18) + 3 -12t + 2t2 = (-108 + 18t) + 3 Reorder the terms: -12t + 2t2 = -108 + 3 + 18t Combine like terms: -108 + 3 = -105 -12t + 2t2 = -105 + 18t Solving -12t + 2t2 = -105 + 18t Solving for variable 't'. Reorder the terms: 105 + -12t + -18t + 2t2 = -105 + 18t + 105 + -18t Combine like terms: -12t + -18t = -30t 105 + -30t + 2t2 = -105 + 18t + 105 + -18t Reorder the terms: 105 + -30t + 2t2 = -105 + 105 + 18t + -18t Combine like terms: -105 + 105 = 0 105 + -30t + 2t2 = 0 + 18t + -18t 105 + -30t + 2t2 = 18t + -18t Combine like terms: 18t + -18t = 0 105 + -30t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 52.5 + -15t + t2 = 0 Move the constant term to the right: Add '-52.5' to each side of the equation. 52.5 + -15t + -52.5 + t2 = 0 + -52.5 Reorder the terms: 52.5 + -52.5 + -15t + t2 = 0 + -52.5 Combine like terms: 52.5 + -52.5 = 0.0 0.0 + -15t + t2 = 0 + -52.5 -15t + t2 = 0 + -52.5 Combine like terms: 0 + -52.5 = -52.5 -15t + t2 = -52.5 The t term is -15t. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15t + 56.25 + t2 = -52.5 + 56.25 Reorder the terms: 56.25 + -15t + t2 = -52.5 + 56.25 Combine like terms: -52.5 + 56.25 = 3.75 56.25 + -15t + t2 = 3.75 Factor a perfect square on the left side: (t + -7.5)(t + -7.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (t + -7.5) equal to 1.936491673 and -1.936491673.Subproblem 1
t + -7.5 = 1.936491673 Simplifying t + -7.5 = 1.936491673 Reorder the terms: -7.5 + t = 1.936491673 Solving -7.5 + t = 1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = 1.936491673 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = 1.936491673 + 7.5 t = 1.936491673 + 7.5 Combine like terms: 1.936491673 + 7.5 = 9.436491673 t = 9.436491673 Simplifying t = 9.436491673Subproblem 2
t + -7.5 = -1.936491673 Simplifying t + -7.5 = -1.936491673 Reorder the terms: -7.5 + t = -1.936491673 Solving -7.5 + t = -1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = -1.936491673 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = -1.936491673 + 7.5 t = -1.936491673 + 7.5 Combine like terms: -1.936491673 + 7.5 = 5.563508327 t = 5.563508327 Simplifying t = 5.563508327Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.436491673, 5.563508327}
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