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Simplifying -3(t + -5) * 6t = 5t + -5 Reorder the terms: -3(-5 + t) * 6t = 5t + -5 Reorder the terms for easier multiplication: -3 * 6t(-5 + t) = 5t + -5 Multiply -3 * 6 -18t(-5 + t) = 5t + -5 (-5 * -18t + t * -18t) = 5t + -5 (90t + -18t2) = 5t + -5 Reorder the terms: 90t + -18t2 = -5 + 5t Solving 90t + -18t2 = -5 + 5t Solving for variable 't'. Reorder the terms: 5 + 90t + -5t + -18t2 = -5 + 5t + 5 + -5t Combine like terms: 90t + -5t = 85t 5 + 85t + -18t2 = -5 + 5t + 5 + -5t Reorder the terms: 5 + 85t + -18t2 = -5 + 5 + 5t + -5t Combine like terms: -5 + 5 = 0 5 + 85t + -18t2 = 0 + 5t + -5t 5 + 85t + -18t2 = 5t + -5t Combine like terms: 5t + -5t = 0 5 + 85t + -18t2 = 0 Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. -0.2777777778 + -4.722222222t + t2 = 0 Move the constant term to the right: Add '0.2777777778' to each side of the equation. -0.2777777778 + -4.722222222t + 0.2777777778 + t2 = 0 + 0.2777777778 Reorder the terms: -0.2777777778 + 0.2777777778 + -4.722222222t + t2 = 0 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + -4.722222222t + t2 = 0 + 0.2777777778 -4.722222222t + t2 = 0 + 0.2777777778 Combine like terms: 0 + 0.2777777778 = 0.2777777778 -4.722222222t + t2 = 0.2777777778 The t term is -4.722222222t. Take half its coefficient (-2.361111111). Square it (5.574845678) and add it to both sides. Add '5.574845678' to each side of the equation. -4.722222222t + 5.574845678 + t2 = 0.2777777778 + 5.574845678 Reorder the terms: 5.574845678 + -4.722222222t + t2 = 0.2777777778 + 5.574845678 Combine like terms: 0.2777777778 + 5.574845678 = 5.8526234558 5.574845678 + -4.722222222t + t2 = 5.8526234558 Factor a perfect square on the left side: (t + -2.361111111)(t + -2.361111111) = 5.8526234558 Calculate the square root of the right side: 2.419219596 Break this problem into two subproblems by setting (t + -2.361111111) equal to 2.419219596 and -2.419219596.Subproblem 1
t + -2.361111111 = 2.419219596 Simplifying t + -2.361111111 = 2.419219596 Reorder the terms: -2.361111111 + t = 2.419219596 Solving -2.361111111 + t = 2.419219596 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.361111111' to each side of the equation. -2.361111111 + 2.361111111 + t = 2.419219596 + 2.361111111 Combine like terms: -2.361111111 + 2.361111111 = 0.000000000 0.000000000 + t = 2.419219596 + 2.361111111 t = 2.419219596 + 2.361111111 Combine like terms: 2.419219596 + 2.361111111 = 4.780330707 t = 4.780330707 Simplifying t = 4.780330707Subproblem 2
t + -2.361111111 = -2.419219596 Simplifying t + -2.361111111 = -2.419219596 Reorder the terms: -2.361111111 + t = -2.419219596 Solving -2.361111111 + t = -2.419219596 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.361111111' to each side of the equation. -2.361111111 + 2.361111111 + t = -2.419219596 + 2.361111111 Combine like terms: -2.361111111 + 2.361111111 = 0.000000000 0.000000000 + t = -2.419219596 + 2.361111111 t = -2.419219596 + 2.361111111 Combine like terms: -2.419219596 + 2.361111111 = -0.058108485 t = -0.058108485 Simplifying t = -0.058108485Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.780330707, -0.058108485}
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