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Simplifying -16t2 + 80t = 5 Reorder the terms: 80t + -16t2 = 5 Solving 80t + -16t2 = 5 Solving for variable 't'. Reorder the terms: -5 + 80t + -16t2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 80t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.3125 + -5t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + -5t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + -5t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + -5t + t2 = 0 + -0.3125 -5t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 -5t + t2 = -0.3125 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = -0.3125 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -0.3125 + 6.25 Combine like terms: -0.3125 + 6.25 = 5.9375 6.25 + -5t + t2 = 5.9375 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 5.9375 Calculate the square root of the right side: 2.436698586 Break this problem into two subproblems by setting (t + -2.5) equal to 2.436698586 and -2.436698586.Subproblem 1
t + -2.5 = 2.436698586 Simplifying t + -2.5 = 2.436698586 Reorder the terms: -2.5 + t = 2.436698586 Solving -2.5 + t = 2.436698586 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 2.436698586 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.436698586 + 2.5 t = 2.436698586 + 2.5 Combine like terms: 2.436698586 + 2.5 = 4.936698586 t = 4.936698586 Simplifying t = 4.936698586Subproblem 2
t + -2.5 = -2.436698586 Simplifying t + -2.5 = -2.436698586 Reorder the terms: -2.5 + t = -2.436698586 Solving -2.5 + t = -2.436698586 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -2.436698586 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.436698586 + 2.5 t = -2.436698586 + 2.5 Combine like terms: -2.436698586 + 2.5 = 0.063301414 t = 0.063301414 Simplifying t = 0.063301414Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.936698586, 0.063301414}
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