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Simplifying 0 = 16t2 + 82t + 20 Reorder the terms: 0 = 20 + 82t + 16t2 Solving 0 = 20 + 82t + 16t2 Solving for variable 't'. Combine like terms: 0 + -20 = -20 -20 + -82t + -16t2 = 20 + 82t + 16t2 + -20 + -82t + -16t2 Reorder the terms: -20 + -82t + -16t2 = 20 + -20 + 82t + -82t + 16t2 + -16t2 Combine like terms: 20 + -20 = 0 -20 + -82t + -16t2 = 0 + 82t + -82t + 16t2 + -16t2 -20 + -82t + -16t2 = 82t + -82t + 16t2 + -16t2 Combine like terms: 82t + -82t = 0 -20 + -82t + -16t2 = 0 + 16t2 + -16t2 -20 + -82t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -20 + -82t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-2'. -2(10 + 41t + 8t2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(10 + 41t + 8t2)' equal to zero and attempt to solve: Simplifying 10 + 41t + 8t2 = 0 Solving 10 + 41t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.25 + 5.125t + t2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + 5.125t + -1.25 + t2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + 5.125t + t2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + 5.125t + t2 = 0 + -1.25 5.125t + t2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 5.125t + t2 = -1.25 The t term is 5.125t. Take half its coefficient (2.5625). Square it (6.56640625) and add it to both sides. Add '6.56640625' to each side of the equation. 5.125t + 6.56640625 + t2 = -1.25 + 6.56640625 Reorder the terms: 6.56640625 + 5.125t + t2 = -1.25 + 6.56640625 Combine like terms: -1.25 + 6.56640625 = 5.31640625 6.56640625 + 5.125t + t2 = 5.31640625 Factor a perfect square on the left side: (t + 2.5625)(t + 2.5625) = 5.31640625 Calculate the square root of the right side: 2.305733343 Break this problem into two subproblems by setting (t + 2.5625) equal to 2.305733343 and -2.305733343.Subproblem 1
t + 2.5625 = 2.305733343 Simplifying t + 2.5625 = 2.305733343 Reorder the terms: 2.5625 + t = 2.305733343 Solving 2.5625 + t = 2.305733343 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + t = 2.305733343 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + t = 2.305733343 + -2.5625 t = 2.305733343 + -2.5625 Combine like terms: 2.305733343 + -2.5625 = -0.256766657 t = -0.256766657 Simplifying t = -0.256766657Subproblem 2
t + 2.5625 = -2.305733343 Simplifying t + 2.5625 = -2.305733343 Reorder the terms: 2.5625 + t = -2.305733343 Solving 2.5625 + t = -2.305733343 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + t = -2.305733343 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + t = -2.305733343 + -2.5625 t = -2.305733343 + -2.5625 Combine like terms: -2.305733343 + -2.5625 = -4.868233343 t = -4.868233343 Simplifying t = -4.868233343Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.256766657, -4.868233343}Solution
t = {-0.256766657, -4.868233343}
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