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Simplifying 35 = -16t2 + 42t + 82 Reorder the terms: 35 = 82 + 42t + -16t2 Solving 35 = 82 + 42t + -16t2 Solving for variable 't'. Combine like terms: 35 + -82 = -47 -47 + -42t + 16t2 = 82 + 42t + -16t2 + -82 + -42t + 16t2 Reorder the terms: -47 + -42t + 16t2 = 82 + -82 + 42t + -42t + -16t2 + 16t2 Combine like terms: 82 + -82 = 0 -47 + -42t + 16t2 = 0 + 42t + -42t + -16t2 + 16t2 -47 + -42t + 16t2 = 42t + -42t + -16t2 + 16t2 Combine like terms: 42t + -42t = 0 -47 + -42t + 16t2 = 0 + -16t2 + 16t2 -47 + -42t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -47 + -42t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.9375 + -2.625t + t2 = 0 Move the constant term to the right: Add '2.9375' to each side of the equation. -2.9375 + -2.625t + 2.9375 + t2 = 0 + 2.9375 Reorder the terms: -2.9375 + 2.9375 + -2.625t + t2 = 0 + 2.9375 Combine like terms: -2.9375 + 2.9375 = 0.0000 0.0000 + -2.625t + t2 = 0 + 2.9375 -2.625t + t2 = 0 + 2.9375 Combine like terms: 0 + 2.9375 = 2.9375 -2.625t + t2 = 2.9375 The t term is -2.625t. Take half its coefficient (-1.3125). Square it (1.72265625) and add it to both sides. Add '1.72265625' to each side of the equation. -2.625t + 1.72265625 + t2 = 2.9375 + 1.72265625 Reorder the terms: 1.72265625 + -2.625t + t2 = 2.9375 + 1.72265625 Combine like terms: 2.9375 + 1.72265625 = 4.66015625 1.72265625 + -2.625t + t2 = 4.66015625 Factor a perfect square on the left side: (t + -1.3125)(t + -1.3125) = 4.66015625 Calculate the square root of the right side: 2.158739505 Break this problem into two subproblems by setting (t + -1.3125) equal to 2.158739505 and -2.158739505.Subproblem 1
t + -1.3125 = 2.158739505 Simplifying t + -1.3125 = 2.158739505 Reorder the terms: -1.3125 + t = 2.158739505 Solving -1.3125 + t = 2.158739505 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = 2.158739505 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = 2.158739505 + 1.3125 t = 2.158739505 + 1.3125 Combine like terms: 2.158739505 + 1.3125 = 3.471239505 t = 3.471239505 Simplifying t = 3.471239505Subproblem 2
t + -1.3125 = -2.158739505 Simplifying t + -1.3125 = -2.158739505 Reorder the terms: -1.3125 + t = -2.158739505 Solving -1.3125 + t = -2.158739505 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = -2.158739505 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = -2.158739505 + 1.3125 t = -2.158739505 + 1.3125 Combine like terms: -2.158739505 + 1.3125 = -0.846239505 t = -0.846239505 Simplifying t = -0.846239505Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.471239505, -0.846239505}
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