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Simplifying -16t2 + 45.5t + 5 = 0 Reorder the terms: 5 + 45.5t + -16t2 = 0 Solving 5 + 45.5t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.3125 + -2.84375t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -2.84375t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -2.84375t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -2.84375t + t2 = 0 + 0.3125 -2.84375t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -2.84375t + t2 = 0.3125 The t term is -2.84375t. Take half its coefficient (-1.421875). Square it (2.021728516) and add it to both sides. Add '2.021728516' to each side of the equation. -2.84375t + 2.021728516 + t2 = 0.3125 + 2.021728516 Reorder the terms: 2.021728516 + -2.84375t + t2 = 0.3125 + 2.021728516 Combine like terms: 0.3125 + 2.021728516 = 2.334228516 2.021728516 + -2.84375t + t2 = 2.334228516 Factor a perfect square on the left side: (t + -1.421875)(t + -1.421875) = 2.334228516 Calculate the square root of the right side: 1.527818221 Break this problem into two subproblems by setting (t + -1.421875) equal to 1.527818221 and -1.527818221.Subproblem 1
t + -1.421875 = 1.527818221 Simplifying t + -1.421875 = 1.527818221 Reorder the terms: -1.421875 + t = 1.527818221 Solving -1.421875 + t = 1.527818221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.421875' to each side of the equation. -1.421875 + 1.421875 + t = 1.527818221 + 1.421875 Combine like terms: -1.421875 + 1.421875 = 0.000000 0.000000 + t = 1.527818221 + 1.421875 t = 1.527818221 + 1.421875 Combine like terms: 1.527818221 + 1.421875 = 2.949693221 t = 2.949693221 Simplifying t = 2.949693221Subproblem 2
t + -1.421875 = -1.527818221 Simplifying t + -1.421875 = -1.527818221 Reorder the terms: -1.421875 + t = -1.527818221 Solving -1.421875 + t = -1.527818221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.421875' to each side of the equation. -1.421875 + 1.421875 + t = -1.527818221 + 1.421875 Combine like terms: -1.421875 + 1.421875 = 0.000000 0.000000 + t = -1.527818221 + 1.421875 t = -1.527818221 + 1.421875 Combine like terms: -1.527818221 + 1.421875 = -0.105943221 t = -0.105943221 Simplifying t = -0.105943221Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.949693221, -0.105943221}
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