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Simplifying 3t(t + -1) = 1 Reorder the terms: 3t(-1 + t) = 1 (-1 * 3t + t * 3t) = 1 (-3t + 3t2) = 1 Solving -3t + 3t2 = 1 Solving for variable 't'. Reorder the terms: -1 + -3t + 3t2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -3t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + -1t + t2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -1t + 0.3333333333 + t2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -1t + t2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -1t + t2 = 0 + 0.3333333333 -1t + t2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -1t + t2 = 0.3333333333 The t term is -1t. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1t + 0.25 + t2 = 0.3333333333 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 0.25 + -1t + t2 = 0.5833333333 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (t + -0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
t + -0.5 = 0.763762616 Simplifying t + -0.5 = 0.763762616 Reorder the terms: -0.5 + t = 0.763762616 Solving -0.5 + t = 0.763762616 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = 0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 0.763762616 + 0.5 t = 0.763762616 + 0.5 Combine like terms: 0.763762616 + 0.5 = 1.263762616 t = 1.263762616 Simplifying t = 1.263762616Subproblem 2
t + -0.5 = -0.763762616 Simplifying t + -0.5 = -0.763762616 Reorder the terms: -0.5 + t = -0.763762616 Solving -0.5 + t = -0.763762616 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = -0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -0.763762616 + 0.5 t = -0.763762616 + 0.5 Combine like terms: -0.763762616 + 0.5 = -0.263762616 t = -0.263762616 Simplifying t = -0.263762616Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.263762616, -0.263762616}
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