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Simplifying 5t(7 + -1t) = 4t + 2(9 + t) (7 * 5t + -1t * 5t) = 4t + 2(9 + t) (35t + -5t2) = 4t + 2(9 + t) 35t + -5t2 = 4t + (9 * 2 + t * 2) 35t + -5t2 = 4t + (18 + 2t) Reorder the terms: 35t + -5t2 = 18 + 4t + 2t Combine like terms: 4t + 2t = 6t 35t + -5t2 = 18 + 6t Solving 35t + -5t2 = 18 + 6t Solving for variable 't'. Reorder the terms: -18 + 35t + -6t + -5t2 = 18 + 6t + -18 + -6t Combine like terms: 35t + -6t = 29t -18 + 29t + -5t2 = 18 + 6t + -18 + -6t Reorder the terms: -18 + 29t + -5t2 = 18 + -18 + 6t + -6t Combine like terms: 18 + -18 = 0 -18 + 29t + -5t2 = 0 + 6t + -6t -18 + 29t + -5t2 = 6t + -6t Combine like terms: 6t + -6t = 0 -18 + 29t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 3.6 + -5.8t + t2 = 0 Move the constant term to the right: Add '-3.6' to each side of the equation. 3.6 + -5.8t + -3.6 + t2 = 0 + -3.6 Reorder the terms: 3.6 + -3.6 + -5.8t + t2 = 0 + -3.6 Combine like terms: 3.6 + -3.6 = 0.0 0.0 + -5.8t + t2 = 0 + -3.6 -5.8t + t2 = 0 + -3.6 Combine like terms: 0 + -3.6 = -3.6 -5.8t + t2 = -3.6 The t term is -5.8t. Take half its coefficient (-2.9). Square it (8.41) and add it to both sides. Add '8.41' to each side of the equation. -5.8t + 8.41 + t2 = -3.6 + 8.41 Reorder the terms: 8.41 + -5.8t + t2 = -3.6 + 8.41 Combine like terms: -3.6 + 8.41 = 4.81 8.41 + -5.8t + t2 = 4.81 Factor a perfect square on the left side: (t + -2.9)(t + -2.9) = 4.81 Calculate the square root of the right side: 2.19317122 Break this problem into two subproblems by setting (t + -2.9) equal to 2.19317122 and -2.19317122.Subproblem 1
t + -2.9 = 2.19317122 Simplifying t + -2.9 = 2.19317122 Reorder the terms: -2.9 + t = 2.19317122 Solving -2.9 + t = 2.19317122 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.9' to each side of the equation. -2.9 + 2.9 + t = 2.19317122 + 2.9 Combine like terms: -2.9 + 2.9 = 0.0 0.0 + t = 2.19317122 + 2.9 t = 2.19317122 + 2.9 Combine like terms: 2.19317122 + 2.9 = 5.09317122 t = 5.09317122 Simplifying t = 5.09317122Subproblem 2
t + -2.9 = -2.19317122 Simplifying t + -2.9 = -2.19317122 Reorder the terms: -2.9 + t = -2.19317122 Solving -2.9 + t = -2.19317122 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.9' to each side of the equation. -2.9 + 2.9 + t = -2.19317122 + 2.9 Combine like terms: -2.9 + 2.9 = 0.0 0.0 + t = -2.19317122 + 2.9 t = -2.19317122 + 2.9 Combine like terms: -2.19317122 + 2.9 = 0.70682878 t = 0.70682878 Simplifying t = 0.70682878Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.09317122, 0.70682878}
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