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Simplifying t = -16t2 + 15t + 5 Reorder the terms: t = 5 + 15t + -16t2 Solving t = 5 + 15t + -16t2 Solving for variable 't'. Reorder the terms: -5 + t + -15t + 16t2 = 5 + 15t + -16t2 + -5 + -15t + 16t2 Combine like terms: t + -15t = -14t -5 + -14t + 16t2 = 5 + 15t + -16t2 + -5 + -15t + 16t2 Reorder the terms: -5 + -14t + 16t2 = 5 + -5 + 15t + -15t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -14t + 16t2 = 0 + 15t + -15t + -16t2 + 16t2 -5 + -14t + 16t2 = 15t + -15t + -16t2 + 16t2 Combine like terms: 15t + -15t = 0 -5 + -14t + 16t2 = 0 + -16t2 + 16t2 -5 + -14t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -14t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -0.875t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -0.875t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -0.875t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -0.875t + t2 = 0 + 0.3125 -0.875t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -0.875t + t2 = 0.3125 The t term is -0.875t. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875t + 0.19140625 + t2 = 0.3125 + 0.19140625 Reorder the terms: 0.19140625 + -0.875t + t2 = 0.3125 + 0.19140625 Combine like terms: 0.3125 + 0.19140625 = 0.50390625 0.19140625 + -0.875t + t2 = 0.50390625 Factor a perfect square on the left side: (t + -0.4375)(t + -0.4375) = 0.50390625 Calculate the square root of the right side: 0.709863543 Break this problem into two subproblems by setting (t + -0.4375) equal to 0.709863543 and -0.709863543.Subproblem 1
t + -0.4375 = 0.709863543 Simplifying t + -0.4375 = 0.709863543 Reorder the terms: -0.4375 + t = 0.709863543 Solving -0.4375 + t = 0.709863543 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + t = 0.709863543 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + t = 0.709863543 + 0.4375 t = 0.709863543 + 0.4375 Combine like terms: 0.709863543 + 0.4375 = 1.147363543 t = 1.147363543 Simplifying t = 1.147363543Subproblem 2
t + -0.4375 = -0.709863543 Simplifying t + -0.4375 = -0.709863543 Reorder the terms: -0.4375 + t = -0.709863543 Solving -0.4375 + t = -0.709863543 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + t = -0.709863543 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + t = -0.709863543 + 0.4375 t = -0.709863543 + 0.4375 Combine like terms: -0.709863543 + 0.4375 = -0.272363543 t = -0.272363543 Simplifying t = -0.272363543Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.147363543, -0.272363543}
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