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Simplifying 449 = -16t2 + -4t + 470 Reorder the terms: 449 = 470 + -4t + -16t2 Solving 449 = 470 + -4t + -16t2 Solving for variable 't'. Combine like terms: 449 + -470 = -21 -21 + 4t + 16t2 = 470 + -4t + -16t2 + -470 + 4t + 16t2 Reorder the terms: -21 + 4t + 16t2 = 470 + -470 + -4t + 4t + -16t2 + 16t2 Combine like terms: 470 + -470 = 0 -21 + 4t + 16t2 = 0 + -4t + 4t + -16t2 + 16t2 -21 + 4t + 16t2 = -4t + 4t + -16t2 + 16t2 Combine like terms: -4t + 4t = 0 -21 + 4t + 16t2 = 0 + -16t2 + 16t2 -21 + 4t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -21 + 4t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.3125 + 0.25t + t2 = 0 Move the constant term to the right: Add '1.3125' to each side of the equation. -1.3125 + 0.25t + 1.3125 + t2 = 0 + 1.3125 Reorder the terms: -1.3125 + 1.3125 + 0.25t + t2 = 0 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + 0.25t + t2 = 0 + 1.3125 0.25t + t2 = 0 + 1.3125 Combine like terms: 0 + 1.3125 = 1.3125 0.25t + t2 = 1.3125 The t term is 0.25t. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25t + 0.015625 + t2 = 1.3125 + 0.015625 Reorder the terms: 0.015625 + 0.25t + t2 = 1.3125 + 0.015625 Combine like terms: 1.3125 + 0.015625 = 1.328125 0.015625 + 0.25t + t2 = 1.328125 Factor a perfect square on the left side: (t + 0.125)(t + 0.125) = 1.328125 Calculate the square root of the right side: 1.152443057 Break this problem into two subproblems by setting (t + 0.125) equal to 1.152443057 and -1.152443057.Subproblem 1
t + 0.125 = 1.152443057 Simplifying t + 0.125 = 1.152443057 Reorder the terms: 0.125 + t = 1.152443057 Solving 0.125 + t = 1.152443057 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = 1.152443057 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = 1.152443057 + -0.125 t = 1.152443057 + -0.125 Combine like terms: 1.152443057 + -0.125 = 1.027443057 t = 1.027443057 Simplifying t = 1.027443057Subproblem 2
t + 0.125 = -1.152443057 Simplifying t + 0.125 = -1.152443057 Reorder the terms: 0.125 + t = -1.152443057 Solving 0.125 + t = -1.152443057 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = -1.152443057 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = -1.152443057 + -0.125 t = -1.152443057 + -0.125 Combine like terms: -1.152443057 + -0.125 = -1.277443057 t = -1.277443057 Simplifying t = -1.277443057Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.027443057, -1.277443057}
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