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Simplifying 0 = -16t2 + 83t + 12389 Reorder the terms: 0 = 12389 + 83t + -16t2 Solving 0 = 12389 + 83t + -16t2 Solving for variable 't'. Combine like terms: 0 + -12389 = -12389 -12389 + -83t + 16t2 = 12389 + 83t + -16t2 + -12389 + -83t + 16t2 Reorder the terms: -12389 + -83t + 16t2 = 12389 + -12389 + 83t + -83t + -16t2 + 16t2 Combine like terms: 12389 + -12389 = 0 -12389 + -83t + 16t2 = 0 + 83t + -83t + -16t2 + 16t2 -12389 + -83t + 16t2 = 83t + -83t + -16t2 + 16t2 Combine like terms: 83t + -83t = 0 -12389 + -83t + 16t2 = 0 + -16t2 + 16t2 -12389 + -83t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12389 + -83t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -774.3125 + -5.1875t + t2 = 0 Move the constant term to the right: Add '774.3125' to each side of the equation. -774.3125 + -5.1875t + 774.3125 + t2 = 0 + 774.3125 Reorder the terms: -774.3125 + 774.3125 + -5.1875t + t2 = 0 + 774.3125 Combine like terms: -774.3125 + 774.3125 = 0.0000 0.0000 + -5.1875t + t2 = 0 + 774.3125 -5.1875t + t2 = 0 + 774.3125 Combine like terms: 0 + 774.3125 = 774.3125 -5.1875t + t2 = 774.3125 The t term is -5.1875t. Take half its coefficient (-2.59375). Square it (6.727539063) and add it to both sides. Add '6.727539063' to each side of the equation. -5.1875t + 6.727539063 + t2 = 774.3125 + 6.727539063 Reorder the terms: 6.727539063 + -5.1875t + t2 = 774.3125 + 6.727539063 Combine like terms: 774.3125 + 6.727539063 = 781.040039063 6.727539063 + -5.1875t + t2 = 781.040039063 Factor a perfect square on the left side: (t + -2.59375)(t + -2.59375) = 781.040039063 Calculate the square root of the right side: 27.947093571 Break this problem into two subproblems by setting (t + -2.59375) equal to 27.947093571 and -27.947093571.Subproblem 1
t + -2.59375 = 27.947093571 Simplifying t + -2.59375 = 27.947093571 Reorder the terms: -2.59375 + t = 27.947093571 Solving -2.59375 + t = 27.947093571 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.59375' to each side of the equation. -2.59375 + 2.59375 + t = 27.947093571 + 2.59375 Combine like terms: -2.59375 + 2.59375 = 0.00000 0.00000 + t = 27.947093571 + 2.59375 t = 27.947093571 + 2.59375 Combine like terms: 27.947093571 + 2.59375 = 30.540843571 t = 30.540843571 Simplifying t = 30.540843571Subproblem 2
t + -2.59375 = -27.947093571 Simplifying t + -2.59375 = -27.947093571 Reorder the terms: -2.59375 + t = -27.947093571 Solving -2.59375 + t = -27.947093571 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.59375' to each side of the equation. -2.59375 + 2.59375 + t = -27.947093571 + 2.59375 Combine like terms: -2.59375 + 2.59375 = 0.00000 0.00000 + t = -27.947093571 + 2.59375 t = -27.947093571 + 2.59375 Combine like terms: -27.947093571 + 2.59375 = -25.353343571 t = -25.353343571 Simplifying t = -25.353343571Solution
The solution to the problem is based on the solutions from the subproblems. t = {30.540843571, -25.353343571}
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