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Simplifying 448(t) = -16t2 + 16t + 480 Reorder the terms: 448t = 480 + 16t + -16t2 Solving 448t = 480 + 16t + -16t2 Solving for variable 't'. Reorder the terms: -480 + 448t + -16t + 16t2 = 480 + 16t + -16t2 + -480 + -16t + 16t2 Combine like terms: 448t + -16t = 432t -480 + 432t + 16t2 = 480 + 16t + -16t2 + -480 + -16t + 16t2 Reorder the terms: -480 + 432t + 16t2 = 480 + -480 + 16t + -16t + -16t2 + 16t2 Combine like terms: 480 + -480 = 0 -480 + 432t + 16t2 = 0 + 16t + -16t + -16t2 + 16t2 -480 + 432t + 16t2 = 16t + -16t + -16t2 + 16t2 Combine like terms: 16t + -16t = 0 -480 + 432t + 16t2 = 0 + -16t2 + 16t2 -480 + 432t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -480 + 432t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-30 + 27t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-30 + 27t + t2)' equal to zero and attempt to solve: Simplifying -30 + 27t + t2 = 0 Solving -30 + 27t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '30' to each side of the equation. -30 + 27t + 30 + t2 = 0 + 30 Reorder the terms: -30 + 30 + 27t + t2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 27t + t2 = 0 + 30 27t + t2 = 0 + 30 Combine like terms: 0 + 30 = 30 27t + t2 = 30 The t term is 27t. Take half its coefficient (13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. 27t + 182.25 + t2 = 30 + 182.25 Reorder the terms: 182.25 + 27t + t2 = 30 + 182.25 Combine like terms: 30 + 182.25 = 212.25 182.25 + 27t + t2 = 212.25 Factor a perfect square on the left side: (t + 13.5)(t + 13.5) = 212.25 Calculate the square root of the right side: 14.568802284 Break this problem into two subproblems by setting (t + 13.5) equal to 14.568802284 and -14.568802284.Subproblem 1
t + 13.5 = 14.568802284 Simplifying t + 13.5 = 14.568802284 Reorder the terms: 13.5 + t = 14.568802284 Solving 13.5 + t = 14.568802284 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = 14.568802284 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = 14.568802284 + -13.5 t = 14.568802284 + -13.5 Combine like terms: 14.568802284 + -13.5 = 1.068802284 t = 1.068802284 Simplifying t = 1.068802284Subproblem 2
t + 13.5 = -14.568802284 Simplifying t + 13.5 = -14.568802284 Reorder the terms: 13.5 + t = -14.568802284 Solving 13.5 + t = -14.568802284 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = -14.568802284 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = -14.568802284 + -13.5 t = -14.568802284 + -13.5 Combine like terms: -14.568802284 + -13.5 = -28.068802284 t = -28.068802284 Simplifying t = -28.068802284Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.068802284, -28.068802284}Solution
t = {1.068802284, -28.068802284}
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