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Simplifying 32 = -16t2 + 56t Reorder the terms: 32 = 56t + -16t2 Solving 32 = 56t + -16t2 Solving for variable 't'. Reorder the terms: 32 + -56t + 16t2 = 56t + -56t + -16t2 + 16t2 Combine like terms: 56t + -56t = 0 32 + -56t + 16t2 = 0 + -16t2 + 16t2 32 + -56t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 32 + -56t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(4 + -7t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(4 + -7t + 2t2)' equal to zero and attempt to solve: Simplifying 4 + -7t + 2t2 = 0 Solving 4 + -7t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2 + -3.5t + t2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -3.5t + -2 + t2 = 0 + -2 Reorder the terms: 2 + -2 + -3.5t + t2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -3.5t + t2 = 0 + -2 -3.5t + t2 = 0 + -2 Combine like terms: 0 + -2 = -2 -3.5t + t2 = -2 The t term is -3.5t. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5t + 3.0625 + t2 = -2 + 3.0625 Reorder the terms: 3.0625 + -3.5t + t2 = -2 + 3.0625 Combine like terms: -2 + 3.0625 = 1.0625 3.0625 + -3.5t + t2 = 1.0625 Factor a perfect square on the left side: (t + -1.75)(t + -1.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (t + -1.75) equal to 1.030776406 and -1.030776406.Subproblem 1
t + -1.75 = 1.030776406 Simplifying t + -1.75 = 1.030776406 Reorder the terms: -1.75 + t = 1.030776406 Solving -1.75 + t = 1.030776406 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + t = 1.030776406 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + t = 1.030776406 + 1.75 t = 1.030776406 + 1.75 Combine like terms: 1.030776406 + 1.75 = 2.780776406 t = 2.780776406 Simplifying t = 2.780776406Subproblem 2
t + -1.75 = -1.030776406 Simplifying t + -1.75 = -1.030776406 Reorder the terms: -1.75 + t = -1.030776406 Solving -1.75 + t = -1.030776406 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + t = -1.030776406 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + t = -1.030776406 + 1.75 t = -1.030776406 + 1.75 Combine like terms: -1.030776406 + 1.75 = 0.719223594 t = 0.719223594 Simplifying t = 0.719223594Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.780776406, 0.719223594}Solution
t = {2.780776406, 0.719223594}
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