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Simplifying 10 = -16t2 + 22t + 5 Reorder the terms: 10 = 5 + 22t + -16t2 Solving 10 = 5 + 22t + -16t2 Solving for variable 't'. Combine like terms: 10 + -5 = 5 5 + -22t + 16t2 = 5 + 22t + -16t2 + -5 + -22t + 16t2 Reorder the terms: 5 + -22t + 16t2 = 5 + -5 + 22t + -22t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 5 + -22t + 16t2 = 0 + 22t + -22t + -16t2 + 16t2 5 + -22t + 16t2 = 22t + -22t + -16t2 + 16t2 Combine like terms: 22t + -22t = 0 5 + -22t + 16t2 = 0 + -16t2 + 16t2 5 + -22t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 5 + -22t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + -1.375t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + -1.375t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + -1.375t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + -1.375t + t2 = 0 + -0.3125 -1.375t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 -1.375t + t2 = -0.3125 The t term is -1.375t. Take half its coefficient (-0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. -1.375t + 0.47265625 + t2 = -0.3125 + 0.47265625 Reorder the terms: 0.47265625 + -1.375t + t2 = -0.3125 + 0.47265625 Combine like terms: -0.3125 + 0.47265625 = 0.16015625 0.47265625 + -1.375t + t2 = 0.16015625 Factor a perfect square on the left side: (t + -0.6875)(t + -0.6875) = 0.16015625 Calculate the square root of the right side: 0.400195265 Break this problem into two subproblems by setting (t + -0.6875) equal to 0.400195265 and -0.400195265.Subproblem 1
t + -0.6875 = 0.400195265 Simplifying t + -0.6875 = 0.400195265 Reorder the terms: -0.6875 + t = 0.400195265 Solving -0.6875 + t = 0.400195265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + t = 0.400195265 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = 0.400195265 + 0.6875 t = 0.400195265 + 0.6875 Combine like terms: 0.400195265 + 0.6875 = 1.087695265 t = 1.087695265 Simplifying t = 1.087695265Subproblem 2
t + -0.6875 = -0.400195265 Simplifying t + -0.6875 = -0.400195265 Reorder the terms: -0.6875 + t = -0.400195265 Solving -0.6875 + t = -0.400195265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + t = -0.400195265 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = -0.400195265 + 0.6875 t = -0.400195265 + 0.6875 Combine like terms: -0.400195265 + 0.6875 = 0.287304735 t = 0.287304735 Simplifying t = 0.287304735Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.087695265, 0.287304735}
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