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Simplifying 0 = -16t2 + -10t + 105 Reorder the terms: 0 = 105 + -10t + -16t2 Solving 0 = 105 + -10t + -16t2 Solving for variable 't'. Combine like terms: 0 + -105 = -105 -105 + 10t + 16t2 = 105 + -10t + -16t2 + -105 + 10t + 16t2 Reorder the terms: -105 + 10t + 16t2 = 105 + -105 + -10t + 10t + -16t2 + 16t2 Combine like terms: 105 + -105 = 0 -105 + 10t + 16t2 = 0 + -10t + 10t + -16t2 + 16t2 -105 + 10t + 16t2 = -10t + 10t + -16t2 + 16t2 Combine like terms: -10t + 10t = 0 -105 + 10t + 16t2 = 0 + -16t2 + 16t2 -105 + 10t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -105 + 10t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -6.5625 + 0.625t + t2 = 0 Move the constant term to the right: Add '6.5625' to each side of the equation. -6.5625 + 0.625t + 6.5625 + t2 = 0 + 6.5625 Reorder the terms: -6.5625 + 6.5625 + 0.625t + t2 = 0 + 6.5625 Combine like terms: -6.5625 + 6.5625 = 0.0000 0.0000 + 0.625t + t2 = 0 + 6.5625 0.625t + t2 = 0 + 6.5625 Combine like terms: 0 + 6.5625 = 6.5625 0.625t + t2 = 6.5625 The t term is 0.625t. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625t + 0.09765625 + t2 = 6.5625 + 0.09765625 Reorder the terms: 0.09765625 + 0.625t + t2 = 6.5625 + 0.09765625 Combine like terms: 6.5625 + 0.09765625 = 6.66015625 0.09765625 + 0.625t + t2 = 6.66015625 Factor a perfect square on the left side: (t + 0.3125)(t + 0.3125) = 6.66015625 Calculate the square root of the right side: 2.580727853 Break this problem into two subproblems by setting (t + 0.3125) equal to 2.580727853 and -2.580727853.Subproblem 1
t + 0.3125 = 2.580727853 Simplifying t + 0.3125 = 2.580727853 Reorder the terms: 0.3125 + t = 2.580727853 Solving 0.3125 + t = 2.580727853 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + t = 2.580727853 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + t = 2.580727853 + -0.3125 t = 2.580727853 + -0.3125 Combine like terms: 2.580727853 + -0.3125 = 2.268227853 t = 2.268227853 Simplifying t = 2.268227853Subproblem 2
t + 0.3125 = -2.580727853 Simplifying t + 0.3125 = -2.580727853 Reorder the terms: 0.3125 + t = -2.580727853 Solving 0.3125 + t = -2.580727853 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + t = -2.580727853 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + t = -2.580727853 + -0.3125 t = -2.580727853 + -0.3125 Combine like terms: -2.580727853 + -0.3125 = -2.893227853 t = -2.893227853 Simplifying t = -2.893227853Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.268227853, -2.893227853}
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