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Simplifying 32 = -16t2 + 32t + 768 Reorder the terms: 32 = 768 + 32t + -16t2 Solving 32 = 768 + 32t + -16t2 Solving for variable 't'. Combine like terms: 32 + -768 = -736 -736 + -32t + 16t2 = 768 + 32t + -16t2 + -768 + -32t + 16t2 Reorder the terms: -736 + -32t + 16t2 = 768 + -768 + 32t + -32t + -16t2 + 16t2 Combine like terms: 768 + -768 = 0 -736 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -736 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -736 + -32t + 16t2 = 0 + -16t2 + 16t2 -736 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -736 + -32t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-46 + -2t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-46 + -2t + t2)' equal to zero and attempt to solve: Simplifying -46 + -2t + t2 = 0 Solving -46 + -2t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '46' to each side of the equation. -46 + -2t + 46 + t2 = 0 + 46 Reorder the terms: -46 + 46 + -2t + t2 = 0 + 46 Combine like terms: -46 + 46 = 0 0 + -2t + t2 = 0 + 46 -2t + t2 = 0 + 46 Combine like terms: 0 + 46 = 46 -2t + t2 = 46 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 46 + 1 Reorder the terms: 1 + -2t + t2 = 46 + 1 Combine like terms: 46 + 1 = 47 1 + -2t + t2 = 47 Factor a perfect square on the left side: (t + -1)(t + -1) = 47 Calculate the square root of the right side: 6.8556546 Break this problem into two subproblems by setting (t + -1) equal to 6.8556546 and -6.8556546.Subproblem 1
t + -1 = 6.8556546 Simplifying t + -1 = 6.8556546 Reorder the terms: -1 + t = 6.8556546 Solving -1 + t = 6.8556546 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 6.8556546 + 1 Combine like terms: -1 + 1 = 0 0 + t = 6.8556546 + 1 t = 6.8556546 + 1 Combine like terms: 6.8556546 + 1 = 7.8556546 t = 7.8556546 Simplifying t = 7.8556546Subproblem 2
t + -1 = -6.8556546 Simplifying t + -1 = -6.8556546 Reorder the terms: -1 + t = -6.8556546 Solving -1 + t = -6.8556546 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -6.8556546 + 1 Combine like terms: -1 + 1 = 0 0 + t = -6.8556546 + 1 t = -6.8556546 + 1 Combine like terms: -6.8556546 + 1 = -5.8556546 t = -5.8556546 Simplifying t = -5.8556546Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.8556546, -5.8556546}Solution
t = {7.8556546, -5.8556546}
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