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Simplifying 600000 = -1718t2 + 82000t Reorder the terms: 600000 = 82000t + -1718t2 Solving 600000 = 82000t + -1718t2 Solving for variable 't'. Reorder the terms: 600000 + -82000t + 1718t2 = 82000t + -82000t + -1718t2 + 1718t2 Combine like terms: 82000t + -82000t = 0 600000 + -82000t + 1718t2 = 0 + -1718t2 + 1718t2 600000 + -82000t + 1718t2 = -1718t2 + 1718t2 Combine like terms: -1718t2 + 1718t2 = 0 600000 + -82000t + 1718t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(300000 + -41000t + 859t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(300000 + -41000t + 859t2)' equal to zero and attempt to solve: Simplifying 300000 + -41000t + 859t2 = 0 Solving 300000 + -41000t + 859t2 = 0 Begin completing the square. Divide all terms by 859 the coefficient of the squared term: Divide each side by '859'. 349.2433062 + -47.72991851t + t2 = 0 Move the constant term to the right: Add '-349.2433062' to each side of the equation. 349.2433062 + -47.72991851t + -349.2433062 + t2 = 0 + -349.2433062 Reorder the terms: 349.2433062 + -349.2433062 + -47.72991851t + t2 = 0 + -349.2433062 Combine like terms: 349.2433062 + -349.2433062 = 0.0000000 0.0000000 + -47.72991851t + t2 = 0 + -349.2433062 -47.72991851t + t2 = 0 + -349.2433062 Combine like terms: 0 + -349.2433062 = -349.2433062 -47.72991851t + t2 = -349.2433062 The t term is -47.72991851t. Take half its coefficient (-23.86495926). Square it (569.5362805) and add it to both sides. Add '569.5362805' to each side of the equation. -47.72991851t + 569.5362805 + t2 = -349.2433062 + 569.5362805 Reorder the terms: 569.5362805 + -47.72991851t + t2 = -349.2433062 + 569.5362805 Combine like terms: -349.2433062 + 569.5362805 = 220.2929743 569.5362805 + -47.72991851t + t2 = 220.2929743 Factor a perfect square on the left side: (t + -23.86495926)(t + -23.86495926) = 220.2929743 Calculate the square root of the right side: 14.84226985 Break this problem into two subproblems by setting (t + -23.86495926) equal to 14.84226985 and -14.84226985.Subproblem 1
t + -23.86495926 = 14.84226985 Simplifying t + -23.86495926 = 14.84226985 Reorder the terms: -23.86495926 + t = 14.84226985 Solving -23.86495926 + t = 14.84226985 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '23.86495926' to each side of the equation. -23.86495926 + 23.86495926 + t = 14.84226985 + 23.86495926 Combine like terms: -23.86495926 + 23.86495926 = 0.00000000 0.00000000 + t = 14.84226985 + 23.86495926 t = 14.84226985 + 23.86495926 Combine like terms: 14.84226985 + 23.86495926 = 38.70722911 t = 38.70722911 Simplifying t = 38.70722911Subproblem 2
t + -23.86495926 = -14.84226985 Simplifying t + -23.86495926 = -14.84226985 Reorder the terms: -23.86495926 + t = -14.84226985 Solving -23.86495926 + t = -14.84226985 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '23.86495926' to each side of the equation. -23.86495926 + 23.86495926 + t = -14.84226985 + 23.86495926 Combine like terms: -23.86495926 + 23.86495926 = 0.00000000 0.00000000 + t = -14.84226985 + 23.86495926 t = -14.84226985 + 23.86495926 Combine like terms: -14.84226985 + 23.86495926 = 9.02268941 t = 9.02268941 Simplifying t = 9.02268941Solution
The solution to the problem is based on the solutions from the subproblems. t = {38.70722911, 9.02268941}Solution
t = {38.70722911, 9.02268941}
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