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Simplifying 2(2t + -5) * t = 20 + -1t Reorder the terms: 2(-5 + 2t) * t = 20 + -1t Reorder the terms for easier multiplication: 2t(-5 + 2t) = 20 + -1t (-5 * 2t + 2t * 2t) = 20 + -1t (-10t + 4t2) = 20 + -1t Solving -10t + 4t2 = 20 + -1t Solving for variable 't'. Reorder the terms: -20 + -10t + t + 4t2 = 20 + -1t + -20 + t Combine like terms: -10t + t = -9t -20 + -9t + 4t2 = 20 + -1t + -20 + t Reorder the terms: -20 + -9t + 4t2 = 20 + -20 + -1t + t Combine like terms: 20 + -20 = 0 -20 + -9t + 4t2 = 0 + -1t + t -20 + -9t + 4t2 = -1t + t Combine like terms: -1t + t = 0 -20 + -9t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5 + -2.25t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.25t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -2.25t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.25t + t2 = 0 + 5 -2.25t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.25t + t2 = 5 The t term is -2.25t. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25t + 1.265625 + t2 = 5 + 1.265625 Reorder the terms: 1.265625 + -2.25t + t2 = 5 + 1.265625 Combine like terms: 5 + 1.265625 = 6.265625 1.265625 + -2.25t + t2 = 6.265625 Factor a perfect square on the left side: (t + -1.125)(t + -1.125) = 6.265625 Calculate the square root of the right side: 2.503123049 Break this problem into two subproblems by setting (t + -1.125) equal to 2.503123049 and -2.503123049.Subproblem 1
t + -1.125 = 2.503123049 Simplifying t + -1.125 = 2.503123049 Reorder the terms: -1.125 + t = 2.503123049 Solving -1.125 + t = 2.503123049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = 2.503123049 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = 2.503123049 + 1.125 t = 2.503123049 + 1.125 Combine like terms: 2.503123049 + 1.125 = 3.628123049 t = 3.628123049 Simplifying t = 3.628123049Subproblem 2
t + -1.125 = -2.503123049 Simplifying t + -1.125 = -2.503123049 Reorder the terms: -1.125 + t = -2.503123049 Solving -1.125 + t = -2.503123049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = -2.503123049 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = -2.503123049 + 1.125 t = -2.503123049 + 1.125 Combine like terms: -2.503123049 + 1.125 = -1.378123049 t = -1.378123049 Simplifying t = -1.378123049Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.628123049, -1.378123049}
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