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Simplifying 2t2 = 5t + -1 Reorder the terms: 2t2 = -1 + 5t Solving 2t2 = -1 + 5t Solving for variable 't'. Reorder the terms: 1 + -5t + 2t2 = -1 + 5t + 1 + -5t Reorder the terms: 1 + -5t + 2t2 = -1 + 1 + 5t + -5t Combine like terms: -1 + 1 = 0 1 + -5t + 2t2 = 0 + 5t + -5t 1 + -5t + 2t2 = 5t + -5t Combine like terms: 5t + -5t = 0 1 + -5t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -2.5t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2.5t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2.5t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2.5t + t2 = 0 + -0.5 -2.5t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2.5t + t2 = -0.5 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = -0.5 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = -0.5 + 1.5625 Combine like terms: -0.5 + 1.5625 = 1.0625 1.5625 + -2.5t + t2 = 1.0625 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (t + -1.25) equal to 1.030776406 and -1.030776406.Subproblem 1
t + -1.25 = 1.030776406 Simplifying t + -1.25 = 1.030776406 Reorder the terms: -1.25 + t = 1.030776406 Solving -1.25 + t = 1.030776406 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.030776406 + 1.25 t = 1.030776406 + 1.25 Combine like terms: 1.030776406 + 1.25 = 2.280776406 t = 2.280776406 Simplifying t = 2.280776406Subproblem 2
t + -1.25 = -1.030776406 Simplifying t + -1.25 = -1.030776406 Reorder the terms: -1.25 + t = -1.030776406 Solving -1.25 + t = -1.030776406 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.030776406 + 1.25 t = -1.030776406 + 1.25 Combine like terms: -1.030776406 + 1.25 = 0.219223594 t = 0.219223594 Simplifying t = 0.219223594Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.280776406, 0.219223594}
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