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Simplifying 0 = -16t2 + 22t + 5 Reorder the terms: 0 = 5 + 22t + -16t2 Solving 0 = 5 + 22t + -16t2 Solving for variable 't'. Combine like terms: 0 + -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 = 0.78515625 0.47265625 + -1.375t + t2 = 0.78515625 Factor a perfect square on the left side: (t + -0.6875)(t + -0.6875) = 0.78515625 Calculate the square root of the right side: 0.88609043 Break this problem into two subproblems by setting (t + -0.6875) equal to 0.88609043 and -0.88609043.Subproblem 1
t + -0.6875 = 0.88609043 Simplifying t + -0.6875 = 0.88609043 Reorder the terms: -0.6875 + t = 0.88609043 Solving -0.6875 + t = 0.88609043 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 = 0.88609043 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = 0.88609043 + 0.6875 t = 0.88609043 + 0.6875 Combine like terms: 0.88609043 + 0.6875 = 1.57359043 t = 1.57359043 Simplifying t = 1.57359043Subproblem 2
t + -0.6875 = -0.88609043 Simplifying t + -0.6875 = -0.88609043 Reorder the terms: -0.6875 + t = -0.88609043 Solving -0.6875 + t = -0.88609043 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 = -0.88609043 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = -0.88609043 + 0.6875 t = -0.88609043 + 0.6875 Combine like terms: -0.88609043 + 0.6875 = -0.19859043 t = -0.19859043 Simplifying t = -0.19859043Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.57359043, -0.19859043}
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