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Simplifying 5 = 16t2 + 24t Reorder the terms: 5 = 24t + 16t2 Solving 5 = 24t + 16t2 Solving for variable 't'. Reorder the terms: 5 + -24t + -16t2 = 24t + -24t + 16t2 + -16t2 Combine like terms: 24t + -24t = 0 5 + -24t + -16t2 = 0 + 16t2 + -16t2 5 + -24t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 5 + -24t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.3125 + 1.5t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + 1.5t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + 1.5t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + 1.5t + t2 = 0 + 0.3125 1.5t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 1.5t + t2 = 0.3125 The t term is 1.5t. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5t + 0.5625 + t2 = 0.3125 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = 0.3125 + 0.5625 Combine like terms: 0.3125 + 0.5625 = 0.875 0.5625 + 1.5t + t2 = 0.875 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 0.875 Calculate the square root of the right side: 0.935414347 Break this problem into two subproblems by setting (t + 0.75) equal to 0.935414347 and -0.935414347.Subproblem 1
t + 0.75 = 0.935414347 Simplifying t + 0.75 = 0.935414347 Reorder the terms: 0.75 + t = 0.935414347 Solving 0.75 + t = 0.935414347 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = 0.935414347 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 0.935414347 + -0.75 t = 0.935414347 + -0.75 Combine like terms: 0.935414347 + -0.75 = 0.185414347 t = 0.185414347 Simplifying t = 0.185414347Subproblem 2
t + 0.75 = -0.935414347 Simplifying t + 0.75 = -0.935414347 Reorder the terms: 0.75 + t = -0.935414347 Solving 0.75 + t = -0.935414347 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = -0.935414347 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -0.935414347 + -0.75 t = -0.935414347 + -0.75 Combine like terms: -0.935414347 + -0.75 = -1.685414347 t = -1.685414347 Simplifying t = -1.685414347Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.185414347, -1.685414347}
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