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Simplifying 0 = -16t2 + 50t + -31 Reorder the terms: 0 = -31 + 50t + -16t2 Solving 0 = -31 + 50t + -16t2 Solving for variable 't'. Combine like terms: 0 + 31 = 31 31 + -50t + 16t2 = -31 + 50t + -16t2 + 31 + -50t + 16t2 Reorder the terms: 31 + -50t + 16t2 = -31 + 31 + 50t + -50t + -16t2 + 16t2 Combine like terms: -31 + 31 = 0 31 + -50t + 16t2 = 0 + 50t + -50t + -16t2 + 16t2 31 + -50t + 16t2 = 50t + -50t + -16t2 + 16t2 Combine like terms: 50t + -50t = 0 31 + -50t + 16t2 = 0 + -16t2 + 16t2 31 + -50t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 31 + -50t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.9375 + -3.125t + t2 = 0 Move the constant term to the right: Add '-1.9375' to each side of the equation. 1.9375 + -3.125t + -1.9375 + t2 = 0 + -1.9375 Reorder the terms: 1.9375 + -1.9375 + -3.125t + t2 = 0 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + -3.125t + t2 = 0 + -1.9375 -3.125t + t2 = 0 + -1.9375 Combine like terms: 0 + -1.9375 = -1.9375 -3.125t + t2 = -1.9375 The t term is -3.125t. Take half its coefficient (-1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. -3.125t + 2.44140625 + t2 = -1.9375 + 2.44140625 Reorder the terms: 2.44140625 + -3.125t + t2 = -1.9375 + 2.44140625 Combine like terms: -1.9375 + 2.44140625 = 0.50390625 2.44140625 + -3.125t + t2 = 0.50390625 Factor a perfect square on the left side: (t + -1.5625)(t + -1.5625) = 0.50390625 Calculate the square root of the right side: 0.709863543 Break this problem into two subproblems by setting (t + -1.5625) equal to 0.709863543 and -0.709863543.Subproblem 1
t + -1.5625 = 0.709863543 Simplifying t + -1.5625 = 0.709863543 Reorder the terms: -1.5625 + t = 0.709863543 Solving -1.5625 + t = 0.709863543 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + t = 0.709863543 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + t = 0.709863543 + 1.5625 t = 0.709863543 + 1.5625 Combine like terms: 0.709863543 + 1.5625 = 2.272363543 t = 2.272363543 Simplifying t = 2.272363543Subproblem 2
t + -1.5625 = -0.709863543 Simplifying t + -1.5625 = -0.709863543 Reorder the terms: -1.5625 + t = -0.709863543 Solving -1.5625 + t = -0.709863543 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + t = -0.709863543 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + t = -0.709863543 + 1.5625 t = -0.709863543 + 1.5625 Combine like terms: -0.709863543 + 1.5625 = 0.852636457 t = 0.852636457 Simplifying t = 0.852636457Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.272363543, 0.852636457}
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