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Simplifying 8 = -16t2 + 120t + 5 Reorder the terms: 8 = 5 + 120t + -16t2 Solving 8 = 5 + 120t + -16t2 Solving for variable 't'. Combine like terms: 8 + -5 = 3 3 + -120t + 16t2 = 5 + 120t + -16t2 + -5 + -120t + 16t2 Reorder the terms: 3 + -120t + 16t2 = 5 + -5 + 120t + -120t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 3 + -120t + 16t2 = 0 + 120t + -120t + -16t2 + 16t2 3 + -120t + 16t2 = 120t + -120t + -16t2 + 16t2 Combine like terms: 120t + -120t = 0 3 + -120t + 16t2 = 0 + -16t2 + 16t2 3 + -120t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 3 + -120t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + -7.5t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -7.5t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -7.5t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -7.5t + t2 = 0 + -0.1875 -7.5t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -7.5t + t2 = -0.1875 The t term is -7.5t. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5t + 14.0625 + t2 = -0.1875 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = -0.1875 + 14.0625 Combine like terms: -0.1875 + 14.0625 = 13.875 14.0625 + -7.5t + t2 = 13.875 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 13.875 Calculate the square root of the right side: 3.724916106 Break this problem into two subproblems by setting (t + -3.75) equal to 3.724916106 and -3.724916106.Subproblem 1
t + -3.75 = 3.724916106 Simplifying t + -3.75 = 3.724916106 Reorder the terms: -3.75 + t = 3.724916106 Solving -3.75 + t = 3.724916106 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = 3.724916106 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 3.724916106 + 3.75 t = 3.724916106 + 3.75 Combine like terms: 3.724916106 + 3.75 = 7.474916106 t = 7.474916106 Simplifying t = 7.474916106Subproblem 2
t + -3.75 = -3.724916106 Simplifying t + -3.75 = -3.724916106 Reorder the terms: -3.75 + t = -3.724916106 Solving -3.75 + t = -3.724916106 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = -3.724916106 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -3.724916106 + 3.75 t = -3.724916106 + 3.75 Combine like terms: -3.724916106 + 3.75 = 0.025083894 t = 0.025083894 Simplifying t = 0.025083894Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.474916106, 0.025083894}
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