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Simplifying -16t2 + -202t + -82 = 0 Reorder the terms: -82 + -202t + -16t2 = 0 Solving -82 + -202t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '-2'. -2(41 + 101t + 8t2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(41 + 101t + 8t2)' equal to zero and attempt to solve: Simplifying 41 + 101t + 8t2 = 0 Solving 41 + 101t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 5.125 + 12.625t + t2 = 0 Move the constant term to the right: Add '-5.125' to each side of the equation. 5.125 + 12.625t + -5.125 + t2 = 0 + -5.125 Reorder the terms: 5.125 + -5.125 + 12.625t + t2 = 0 + -5.125 Combine like terms: 5.125 + -5.125 = 0.000 0.000 + 12.625t + t2 = 0 + -5.125 12.625t + t2 = 0 + -5.125 Combine like terms: 0 + -5.125 = -5.125 12.625t + t2 = -5.125 The t term is 12.625t. Take half its coefficient (6.3125). Square it (39.84765625) and add it to both sides. Add '39.84765625' to each side of the equation. 12.625t + 39.84765625 + t2 = -5.125 + 39.84765625 Reorder the terms: 39.84765625 + 12.625t + t2 = -5.125 + 39.84765625 Combine like terms: -5.125 + 39.84765625 = 34.72265625 39.84765625 + 12.625t + t2 = 34.72265625 Factor a perfect square on the left side: (t + 6.3125)(t + 6.3125) = 34.72265625 Calculate the square root of the right side: 5.892593338 Break this problem into two subproblems by setting (t + 6.3125) equal to 5.892593338 and -5.892593338.Subproblem 1
t + 6.3125 = 5.892593338 Simplifying t + 6.3125 = 5.892593338 Reorder the terms: 6.3125 + t = 5.892593338 Solving 6.3125 + t = 5.892593338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-6.3125' to each side of the equation. 6.3125 + -6.3125 + t = 5.892593338 + -6.3125 Combine like terms: 6.3125 + -6.3125 = 0.0000 0.0000 + t = 5.892593338 + -6.3125 t = 5.892593338 + -6.3125 Combine like terms: 5.892593338 + -6.3125 = -0.419906662 t = -0.419906662 Simplifying t = -0.419906662Subproblem 2
t + 6.3125 = -5.892593338 Simplifying t + 6.3125 = -5.892593338 Reorder the terms: 6.3125 + t = -5.892593338 Solving 6.3125 + t = -5.892593338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-6.3125' to each side of the equation. 6.3125 + -6.3125 + t = -5.892593338 + -6.3125 Combine like terms: 6.3125 + -6.3125 = 0.0000 0.0000 + t = -5.892593338 + -6.3125 t = -5.892593338 + -6.3125 Combine like terms: -5.892593338 + -6.3125 = -12.205093338 t = -12.205093338 Simplifying t = -12.205093338Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.419906662, -12.205093338}Solution
t = {-0.419906662, -12.205093338}
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