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Simplifying 1900 = 0.0089t2 + 1.1149t + 78.4491 Reorder the terms: 1900 = 78.4491 + 1.1149t + 0.0089t2 Solving 1900 = 78.4491 + 1.1149t + 0.0089t2 Solving for variable 't'. Combine like terms: 1900 + -78.4491 = 1821.5509 1821.5509 + -1.1149t + -0.0089t2 = 78.4491 + 1.1149t + 0.0089t2 + -78.4491 + -1.1149t + -0.0089t2 Reorder the terms: 1821.5509 + -1.1149t + -0.0089t2 = 78.4491 + -78.4491 + 1.1149t + -1.1149t + 0.0089t2 + -0.0089t2 Combine like terms: 78.4491 + -78.4491 = 0.0000 1821.5509 + -1.1149t + -0.0089t2 = 0.0000 + 1.1149t + -1.1149t + 0.0089t2 + -0.0089t2 1821.5509 + -1.1149t + -0.0089t2 = 1.1149t + -1.1149t + 0.0089t2 + -0.0089t2 Combine like terms: 1.1149t + -1.1149t = 0.0000 1821.5509 + -1.1149t + -0.0089t2 = 0.0000 + 0.0089t2 + -0.0089t2 1821.5509 + -1.1149t + -0.0089t2 = 0.0089t2 + -0.0089t2 Combine like terms: 0.0089t2 + -0.0089t2 = 0.0000 1821.5509 + -1.1149t + -0.0089t2 = 0.0000 Begin completing the square. Divide all terms by -0.0089 the coefficient of the squared term: Divide each side by '-0.0089'. -204668.6404 + 125.2696629t + t2 = 0 Move the constant term to the right: Add '204668.6404' to each side of the equation. -204668.6404 + 125.2696629t + 204668.6404 + t2 = 0 + 204668.6404 Reorder the terms: -204668.6404 + 204668.6404 + 125.2696629t + t2 = 0 + 204668.6404 Combine like terms: -204668.6404 + 204668.6404 = 0.0000 0.0000 + 125.2696629t + t2 = 0 + 204668.6404 125.2696629t + t2 = 0 + 204668.6404 Combine like terms: 0 + 204668.6404 = 204668.6404 125.2696629t + t2 = 204668.6404 The t term is 125.2696629t. Take half its coefficient (62.63483145). Square it (3923.122111) and add it to both sides. Add '3923.122111' to each side of the equation. 125.2696629t + 3923.122111 + t2 = 204668.6404 + 3923.122111 Reorder the terms: 3923.122111 + 125.2696629t + t2 = 204668.6404 + 3923.122111 Combine like terms: 204668.6404 + 3923.122111 = 208591.762511 3923.122111 + 125.2696629t + t2 = 208591.762511 Factor a perfect square on the left side: (t + 62.63483145)(t + 62.63483145) = 208591.762511 Calculate the square root of the right side: 456.71847183 Break this problem into two subproblems by setting (t + 62.63483145) equal to 456.71847183 and -456.71847183.Subproblem 1
t + 62.63483145 = 456.71847183 Simplifying t + 62.63483145 = 456.71847183 Reorder the terms: 62.63483145 + t = 456.71847183 Solving 62.63483145 + t = 456.71847183 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-62.63483145' to each side of the equation. 62.63483145 + -62.63483145 + t = 456.71847183 + -62.63483145 Combine like terms: 62.63483145 + -62.63483145 = 0.00000000 0.00000000 + t = 456.71847183 + -62.63483145 t = 456.71847183 + -62.63483145 Combine like terms: 456.71847183 + -62.63483145 = 394.08364038 t = 394.08364038 Simplifying t = 394.08364038Subproblem 2
t + 62.63483145 = -456.71847183 Simplifying t + 62.63483145 = -456.71847183 Reorder the terms: 62.63483145 + t = -456.71847183 Solving 62.63483145 + t = -456.71847183 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-62.63483145' to each side of the equation. 62.63483145 + -62.63483145 + t = -456.71847183 + -62.63483145 Combine like terms: 62.63483145 + -62.63483145 = 0.00000000 0.00000000 + t = -456.71847183 + -62.63483145 t = -456.71847183 + -62.63483145 Combine like terms: -456.71847183 + -62.63483145 = -519.35330328 t = -519.35330328 Simplifying t = -519.35330328Solution
The solution to the problem is based on the solutions from the subproblems. t = {394.08364038, -519.35330328}
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