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Simplifying 5t3 + 60t2 = 180t Reorder the terms: 60t2 + 5t3 = 180t Solving 60t2 + 5t3 = 180t Solving for variable 't'. Reorder the terms: -180t + 60t2 + 5t3 = 180t + -180t Combine like terms: 180t + -180t = 0 -180t + 60t2 + 5t3 = 0 Factor out the Greatest Common Factor (GCF), '5t'. 5t(-36 + 12t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor 't' equal to zero and attempt to solve: Simplifying t = 0 Solving t = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t = 0Subproblem 2
Set the factor '(-36 + 12t + t2)' equal to zero and attempt to solve: Simplifying -36 + 12t + t2 = 0 Solving -36 + 12t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '36' to each side of the equation. -36 + 12t + 36 + t2 = 0 + 36 Reorder the terms: -36 + 36 + 12t + t2 = 0 + 36 Combine like terms: -36 + 36 = 0 0 + 12t + t2 = 0 + 36 12t + t2 = 0 + 36 Combine like terms: 0 + 36 = 36 12t + t2 = 36 The t term is 12t. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12t + 36 + t2 = 36 + 36 Reorder the terms: 36 + 12t + t2 = 36 + 36 Combine like terms: 36 + 36 = 72 36 + 12t + t2 = 72 Factor a perfect square on the left side: (t + 6)(t + 6) = 72 Calculate the square root of the right side: 8.485281374 Break this problem into two subproblems by setting (t + 6) equal to 8.485281374 and -8.485281374.Subproblem 1
t + 6 = 8.485281374 Simplifying t + 6 = 8.485281374 Reorder the terms: 6 + t = 8.485281374 Solving 6 + t = 8.485281374 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + t = 8.485281374 + -6 Combine like terms: 6 + -6 = 0 0 + t = 8.485281374 + -6 t = 8.485281374 + -6 Combine like terms: 8.485281374 + -6 = 2.485281374 t = 2.485281374 Simplifying t = 2.485281374Subproblem 2
t + 6 = -8.485281374 Simplifying t + 6 = -8.485281374 Reorder the terms: 6 + t = -8.485281374 Solving 6 + t = -8.485281374 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + t = -8.485281374 + -6 Combine like terms: 6 + -6 = 0 0 + t = -8.485281374 + -6 t = -8.485281374 + -6 Combine like terms: -8.485281374 + -6 = -14.485281374 t = -14.485281374 Simplifying t = -14.485281374Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.485281374, -14.485281374}Solution
t = {0, 2.485281374, -14.485281374}
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