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Simplifying 0 = -16t2 + 82t + 10 Reorder the terms: 0 = 10 + 82t + -16t2 Solving 0 = 10 + 82t + -16t2 Solving for variable 't'. Combine like terms: 0 + -10 = -10 -10 + -82t + 16t2 = 10 + 82t + -16t2 + -10 + -82t + 16t2 Reorder the terms: -10 + -82t + 16t2 = 10 + -10 + 82t + -82t + -16t2 + 16t2 Combine like terms: 10 + -10 = 0 -10 + -82t + 16t2 = 0 + 82t + -82t + -16t2 + 16t2 -10 + -82t + 16t2 = 82t + -82t + -16t2 + 16t2 Combine like terms: 82t + -82t = 0 -10 + -82t + 16t2 = 0 + -16t2 + 16t2 -10 + -82t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -10 + -82t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -41t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -41t + 8t2)' equal to zero and attempt to solve: Simplifying -5 + -41t + 8t2 = 0 Solving -5 + -41t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.625 + -5.125t + t2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + -5.125t + 0.625 + t2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + -5.125t + t2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + -5.125t + t2 = 0 + 0.625 -5.125t + t2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 -5.125t + t2 = 0.625 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 = 0.625 + 6.56640625 Reorder the terms: 6.56640625 + -5.125t + t2 = 0.625 + 6.56640625 Combine like terms: 0.625 + 6.56640625 = 7.19140625 6.56640625 + -5.125t + t2 = 7.19140625 Factor a perfect square on the left side: (t + -2.5625)(t + -2.5625) = 7.19140625 Calculate the square root of the right side: 2.681679744 Break this problem into two subproblems by setting (t + -2.5625) equal to 2.681679744 and -2.681679744.Subproblem 1
t + -2.5625 = 2.681679744 Simplifying t + -2.5625 = 2.681679744 Reorder the terms: -2.5625 + t = 2.681679744 Solving -2.5625 + t = 2.681679744 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 = 2.681679744 + 2.5625 Combine like terms: -2.5625 + 2.5625 = 0.0000 0.0000 + t = 2.681679744 + 2.5625 t = 2.681679744 + 2.5625 Combine like terms: 2.681679744 + 2.5625 = 5.244179744 t = 5.244179744 Simplifying t = 5.244179744Subproblem 2
t + -2.5625 = -2.681679744 Simplifying t + -2.5625 = -2.681679744 Reorder the terms: -2.5625 + t = -2.681679744 Solving -2.5625 + t = -2.681679744 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 = -2.681679744 + 2.5625 Combine like terms: -2.5625 + 2.5625 = 0.0000 0.0000 + t = -2.681679744 + 2.5625 t = -2.681679744 + 2.5625 Combine like terms: -2.681679744 + 2.5625 = -0.119179744 t = -0.119179744 Simplifying t = -0.119179744Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.244179744, -0.119179744}Solution
t = {5.244179744, -0.119179744}
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