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Simplifying 80t = -16t2 + 160t + 80 Reorder the terms: 80t = 80 + 160t + -16t2 Solving 80t = 80 + 160t + -16t2 Solving for variable 't'. Reorder the terms: -80 + 80t + -160t + 16t2 = 80 + 160t + -16t2 + -80 + -160t + 16t2 Combine like terms: 80t + -160t = -80t -80 + -80t + 16t2 = 80 + 160t + -16t2 + -80 + -160t + 16t2 Reorder the terms: -80 + -80t + 16t2 = 80 + -80 + 160t + -160t + -16t2 + 16t2 Combine like terms: 80 + -80 = 0 -80 + -80t + 16t2 = 0 + 160t + -160t + -16t2 + 16t2 -80 + -80t + 16t2 = 160t + -160t + -16t2 + 16t2 Combine like terms: 160t + -160t = 0 -80 + -80t + 16t2 = 0 + -16t2 + 16t2 -80 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -80 + -80t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-5 + -5t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-5 + -5t + t2)' equal to zero and attempt to solve: Simplifying -5 + -5t + t2 = 0 Solving -5 + -5t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -5t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -5t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -5t + t2 = 0 + 5 -5t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -5t + t2 = 5 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 = 5 + 6.25 Reorder the terms: 6.25 + -5t + t2 = 5 + 6.25 Combine like terms: 5 + 6.25 = 11.25 6.25 + -5t + t2 = 11.25 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 11.25 Calculate the square root of the right side: 3.354101966 Break this problem into two subproblems by setting (t + -2.5) equal to 3.354101966 and -3.354101966.Subproblem 1
t + -2.5 = 3.354101966 Simplifying t + -2.5 = 3.354101966 Reorder the terms: -2.5 + t = 3.354101966 Solving -2.5 + t = 3.354101966 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 = 3.354101966 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 3.354101966 + 2.5 t = 3.354101966 + 2.5 Combine like terms: 3.354101966 + 2.5 = 5.854101966 t = 5.854101966 Simplifying t = 5.854101966Subproblem 2
t + -2.5 = -3.354101966 Simplifying t + -2.5 = -3.354101966 Reorder the terms: -2.5 + t = -3.354101966 Solving -2.5 + t = -3.354101966 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 = -3.354101966 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -3.354101966 + 2.5 t = -3.354101966 + 2.5 Combine like terms: -3.354101966 + 2.5 = -0.854101966 t = -0.854101966 Simplifying t = -0.854101966Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.854101966, -0.854101966}Solution
t = {5.854101966, -0.854101966}
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