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Simplifying 303.75 = 27t + -0.3t2 Solving 303.75 = 27t + -0.3t2 Solving for variable 't'. Reorder the terms: 303.75 + -27t + 0.3t2 = 27t + -27t + -0.3t2 + 0.3t2 Combine like terms: 27t + -27t = 0 303.75 + -27t + 0.3t2 = 0 + -0.3t2 + 0.3t2 303.75 + -27t + 0.3t2 = -0.3t2 + 0.3t2 Combine like terms: -0.3t2 + 0.3t2 = 0.0 303.75 + -27t + 0.3t2 = 0.0 Begin completing the square. Divide all terms by 0.3 the coefficient of the squared term: Divide each side by '0.3'. 1012.5 + -90t + t2 = 0 Move the constant term to the right: Add '-1012.5' to each side of the equation. 1012.5 + -90t + -1012.5 + t2 = 0 + -1012.5 Reorder the terms: 1012.5 + -1012.5 + -90t + t2 = 0 + -1012.5 Combine like terms: 1012.5 + -1012.5 = 0.0 0.0 + -90t + t2 = 0 + -1012.5 -90t + t2 = 0 + -1012.5 Combine like terms: 0 + -1012.5 = -1012.5 -90t + t2 = -1012.5 The t term is -90t. Take half its coefficient (-45). Square it (2025) and add it to both sides. Add '2025' to each side of the equation. -90t + 2025 + t2 = -1012.5 + 2025 Reorder the terms: 2025 + -90t + t2 = -1012.5 + 2025 Combine like terms: -1012.5 + 2025 = 1012.5 2025 + -90t + t2 = 1012.5 Factor a perfect square on the left side: (t + -45)(t + -45) = 1012.5 Calculate the square root of the right side: 31.819805153 Break this problem into two subproblems by setting (t + -45) equal to 31.819805153 and -31.819805153.Subproblem 1
t + -45 = 31.819805153 Simplifying t + -45 = 31.819805153 Reorder the terms: -45 + t = 31.819805153 Solving -45 + t = 31.819805153 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + t = 31.819805153 + 45 Combine like terms: -45 + 45 = 0 0 + t = 31.819805153 + 45 t = 31.819805153 + 45 Combine like terms: 31.819805153 + 45 = 76.819805153 t = 76.819805153 Simplifying t = 76.819805153Subproblem 2
t + -45 = -31.819805153 Simplifying t + -45 = -31.819805153 Reorder the terms: -45 + t = -31.819805153 Solving -45 + t = -31.819805153 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + t = -31.819805153 + 45 Combine like terms: -45 + 45 = 0 0 + t = -31.819805153 + 45 t = -31.819805153 + 45 Combine like terms: -31.819805153 + 45 = 13.180194847 t = 13.180194847 Simplifying t = 13.180194847Solution
The solution to the problem is based on the solutions from the subproblems. t = {76.819805153, 13.180194847}
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