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Simplifying 0 = -16t2 + 76t + 46 Reorder the terms: 0 = 46 + 76t + -16t2 Solving 0 = 46 + 76t + -16t2 Solving for variable 't'. Combine like terms: 0 + -46 = -46 -46 + -76t + 16t2 = 46 + 76t + -16t2 + -46 + -76t + 16t2 Reorder the terms: -46 + -76t + 16t2 = 46 + -46 + 76t + -76t + -16t2 + 16t2 Combine like terms: 46 + -46 = 0 -46 + -76t + 16t2 = 0 + 76t + -76t + -16t2 + 16t2 -46 + -76t + 16t2 = 76t + -76t + -16t2 + 16t2 Combine like terms: 76t + -76t = 0 -46 + -76t + 16t2 = 0 + -16t2 + 16t2 -46 + -76t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -46 + -76t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-23 + -38t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-23 + -38t + 8t2)' equal to zero and attempt to solve: Simplifying -23 + -38t + 8t2 = 0 Solving -23 + -38t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.875 + -4.75t + t2 = 0 Move the constant term to the right: Add '2.875' to each side of the equation. -2.875 + -4.75t + 2.875 + t2 = 0 + 2.875 Reorder the terms: -2.875 + 2.875 + -4.75t + t2 = 0 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + -4.75t + t2 = 0 + 2.875 -4.75t + t2 = 0 + 2.875 Combine like terms: 0 + 2.875 = 2.875 -4.75t + t2 = 2.875 The t term is -4.75t. Take half its coefficient (-2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. -4.75t + 5.640625 + t2 = 2.875 + 5.640625 Reorder the terms: 5.640625 + -4.75t + t2 = 2.875 + 5.640625 Combine like terms: 2.875 + 5.640625 = 8.515625 5.640625 + -4.75t + t2 = 8.515625 Factor a perfect square on the left side: (t + -2.375)(t + -2.375) = 8.515625 Calculate the square root of the right side: 2.918154382 Break this problem into two subproblems by setting (t + -2.375) equal to 2.918154382 and -2.918154382.Subproblem 1
t + -2.375 = 2.918154382 Simplifying t + -2.375 = 2.918154382 Reorder the terms: -2.375 + t = 2.918154382 Solving -2.375 + t = 2.918154382 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = 2.918154382 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = 2.918154382 + 2.375 t = 2.918154382 + 2.375 Combine like terms: 2.918154382 + 2.375 = 5.293154382 t = 5.293154382 Simplifying t = 5.293154382Subproblem 2
t + -2.375 = -2.918154382 Simplifying t + -2.375 = -2.918154382 Reorder the terms: -2.375 + t = -2.918154382 Solving -2.375 + t = -2.918154382 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = -2.918154382 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = -2.918154382 + 2.375 t = -2.918154382 + 2.375 Combine like terms: -2.918154382 + 2.375 = -0.543154382 t = -0.543154382 Simplifying t = -0.543154382Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.293154382, -0.543154382}Solution
t = {5.293154382, -0.543154382}
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