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Simplifying 7t(t + 5) = 6t + 5 Reorder the terms: 7t(5 + t) = 6t + 5 (5 * 7t + t * 7t) = 6t + 5 (35t + 7t2) = 6t + 5 Reorder the terms: 35t + 7t2 = 5 + 6t Solving 35t + 7t2 = 5 + 6t Solving for variable 't'. Reorder the terms: -5 + 35t + -6t + 7t2 = 5 + 6t + -5 + -6t Combine like terms: 35t + -6t = 29t -5 + 29t + 7t2 = 5 + 6t + -5 + -6t Reorder the terms: -5 + 29t + 7t2 = 5 + -5 + 6t + -6t Combine like terms: 5 + -5 = 0 -5 + 29t + 7t2 = 0 + 6t + -6t -5 + 29t + 7t2 = 6t + -6t Combine like terms: 6t + -6t = 0 -5 + 29t + 7t2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.7142857143 + 4.142857143t + t2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + 4.142857143t + 0.7142857143 + t2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + 4.142857143t + t2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + 4.142857143t + t2 = 0 + 0.7142857143 4.142857143t + t2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 4.142857143t + t2 = 0.7142857143 The t term is 4.142857143t. Take half its coefficient (2.071428572). Square it (4.290816329) and add it to both sides. Add '4.290816329' to each side of the equation. 4.142857143t + 4.290816329 + t2 = 0.7142857143 + 4.290816329 Reorder the terms: 4.290816329 + 4.142857143t + t2 = 0.7142857143 + 4.290816329 Combine like terms: 0.7142857143 + 4.290816329 = 5.0051020433 4.290816329 + 4.142857143t + t2 = 5.0051020433 Factor a perfect square on the left side: (t + 2.071428572)(t + 2.071428572) = 5.0051020433 Calculate the square root of the right side: 2.237208538 Break this problem into two subproblems by setting (t + 2.071428572) equal to 2.237208538 and -2.237208538.Subproblem 1
t + 2.071428572 = 2.237208538 Simplifying t + 2.071428572 = 2.237208538 Reorder the terms: 2.071428572 + t = 2.237208538 Solving 2.071428572 + t = 2.237208538 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.071428572' to each side of the equation. 2.071428572 + -2.071428572 + t = 2.237208538 + -2.071428572 Combine like terms: 2.071428572 + -2.071428572 = 0.000000000 0.000000000 + t = 2.237208538 + -2.071428572 t = 2.237208538 + -2.071428572 Combine like terms: 2.237208538 + -2.071428572 = 0.165779966 t = 0.165779966 Simplifying t = 0.165779966Subproblem 2
t + 2.071428572 = -2.237208538 Simplifying t + 2.071428572 = -2.237208538 Reorder the terms: 2.071428572 + t = -2.237208538 Solving 2.071428572 + t = -2.237208538 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.071428572' to each side of the equation. 2.071428572 + -2.071428572 + t = -2.237208538 + -2.071428572 Combine like terms: 2.071428572 + -2.071428572 = 0.000000000 0.000000000 + t = -2.237208538 + -2.071428572 t = -2.237208538 + -2.071428572 Combine like terms: -2.237208538 + -2.071428572 = -4.30863711 t = -4.30863711 Simplifying t = -4.30863711Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.165779966, -4.30863711}
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