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Simplifying 2.7t2 + 13.5t + 14 = 6 Reorder the terms: 14 + 13.5t + 2.7t2 = 6 Solving 14 + 13.5t + 2.7t2 = 6 Solving for variable 't'. Reorder the terms: 14 + -6 + 13.5t + 2.7t2 = 6 + -6 Combine like terms: 14 + -6 = 8 8 + 13.5t + 2.7t2 = 6 + -6 Combine like terms: 6 + -6 = 0 8 + 13.5t + 2.7t2 = 0 Begin completing the square. Divide all terms by 2.7 the coefficient of the squared term: Divide each side by '2.7'. 2.962962963 + 5t + t2 = 0 Move the constant term to the right: Add '-2.962962963' to each side of the equation. 2.962962963 + 5t + -2.962962963 + t2 = 0 + -2.962962963 Reorder the terms: 2.962962963 + -2.962962963 + 5t + t2 = 0 + -2.962962963 Combine like terms: 2.962962963 + -2.962962963 = 0.000000000 0.000000000 + 5t + t2 = 0 + -2.962962963 5t + t2 = 0 + -2.962962963 Combine like terms: 0 + -2.962962963 = -2.962962963 5t + t2 = -2.962962963 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = -2.962962963 + 6.25 Reorder the terms: 6.25 + 5t + t2 = -2.962962963 + 6.25 Combine like terms: -2.962962963 + 6.25 = 3.287037037 6.25 + 5t + t2 = 3.287037037 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 3.287037037 Calculate the square root of the right side: 1.813018764 Break this problem into two subproblems by setting (t + 2.5) equal to 1.813018764 and -1.813018764.Subproblem 1
t + 2.5 = 1.813018764 Simplifying t + 2.5 = 1.813018764 Reorder the terms: 2.5 + t = 1.813018764 Solving 2.5 + t = 1.813018764 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 1.813018764 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 1.813018764 + -2.5 t = 1.813018764 + -2.5 Combine like terms: 1.813018764 + -2.5 = -0.686981236 t = -0.686981236 Simplifying t = -0.686981236Subproblem 2
t + 2.5 = -1.813018764 Simplifying t + 2.5 = -1.813018764 Reorder the terms: 2.5 + t = -1.813018764 Solving 2.5 + t = -1.813018764 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -1.813018764 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -1.813018764 + -2.5 t = -1.813018764 + -2.5 Combine like terms: -1.813018764 + -2.5 = -4.313018764 t = -4.313018764 Simplifying t = -4.313018764Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.686981236, -4.313018764}
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