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Simplifying 0 = -16t2 + 160t + 324 Reorder the terms: 0 = 324 + 160t + -16t2 Solving 0 = 324 + 160t + -16t2 Solving for variable 't'. Combine like terms: 0 + -324 = -324 -324 + -160t + 16t2 = 324 + 160t + -16t2 + -324 + -160t + 16t2 Reorder the terms: -324 + -160t + 16t2 = 324 + -324 + 160t + -160t + -16t2 + 16t2 Combine like terms: 324 + -324 = 0 -324 + -160t + 16t2 = 0 + 160t + -160t + -16t2 + 16t2 -324 + -160t + 16t2 = 160t + -160t + -16t2 + 16t2 Combine like terms: 160t + -160t = 0 -324 + -160t + 16t2 = 0 + -16t2 + 16t2 -324 + -160t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -324 + -160t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-81 + -40t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-81 + -40t + 4t2)' equal to zero and attempt to solve: Simplifying -81 + -40t + 4t2 = 0 Solving -81 + -40t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -20.25 + -10t + t2 = 0 Move the constant term to the right: Add '20.25' to each side of the equation. -20.25 + -10t + 20.25 + t2 = 0 + 20.25 Reorder the terms: -20.25 + 20.25 + -10t + t2 = 0 + 20.25 Combine like terms: -20.25 + 20.25 = 0.00 0.00 + -10t + t2 = 0 + 20.25 -10t + t2 = 0 + 20.25 Combine like terms: 0 + 20.25 = 20.25 -10t + t2 = 20.25 The t term is -10t. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10t + 25 + t2 = 20.25 + 25 Reorder the terms: 25 + -10t + t2 = 20.25 + 25 Combine like terms: 20.25 + 25 = 45.25 25 + -10t + t2 = 45.25 Factor a perfect square on the left side: (t + -5)(t + -5) = 45.25 Calculate the square root of the right side: 6.726812024 Break this problem into two subproblems by setting (t + -5) equal to 6.726812024 and -6.726812024.Subproblem 1
t + -5 = 6.726812024 Simplifying t + -5 = 6.726812024 Reorder the terms: -5 + t = 6.726812024 Solving -5 + t = 6.726812024 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = 6.726812024 + 5 Combine like terms: -5 + 5 = 0 0 + t = 6.726812024 + 5 t = 6.726812024 + 5 Combine like terms: 6.726812024 + 5 = 11.726812024 t = 11.726812024 Simplifying t = 11.726812024Subproblem 2
t + -5 = -6.726812024 Simplifying t + -5 = -6.726812024 Reorder the terms: -5 + t = -6.726812024 Solving -5 + t = -6.726812024 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = -6.726812024 + 5 Combine like terms: -5 + 5 = 0 0 + t = -6.726812024 + 5 t = -6.726812024 + 5 Combine like terms: -6.726812024 + 5 = -1.726812024 t = -1.726812024 Simplifying t = -1.726812024Solution
The solution to the problem is based on the solutions from the subproblems. t = {11.726812024, -1.726812024}Solution
t = {11.726812024, -1.726812024}
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