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Simplifying t(12t2 + 60t + -12) = 0 Reorder the terms: t(-12 + 60t + 12t2) = 0 (-12 * t + 60t * t + 12t2 * t) = 0 (-12t + 60t2 + 12t3) = 0 Solving -12t + 60t2 + 12t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '12t'. 12t(-1 + 5t + t2) = 0 Ignore the factor 12.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 '(-1 + 5t + t2)' equal to zero and attempt to solve: Simplifying -1 + 5t + t2 = 0 Solving -1 + 5t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 5t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + 5t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 5t + t2 = 0 + 1 5t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 5t + t2 = 1 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 = 1 + 6.25 Reorder the terms: 6.25 + 5t + t2 = 1 + 6.25 Combine like terms: 1 + 6.25 = 7.25 6.25 + 5t + t2 = 7.25 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (t + 2.5) equal to 2.692582404 and -2.692582404.Subproblem 1
t + 2.5 = 2.692582404 Simplifying t + 2.5 = 2.692582404 Reorder the terms: 2.5 + t = 2.692582404 Solving 2.5 + t = 2.692582404 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 = 2.692582404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 2.692582404 + -2.5 t = 2.692582404 + -2.5 Combine like terms: 2.692582404 + -2.5 = 0.192582404 t = 0.192582404 Simplifying t = 0.192582404Subproblem 2
t + 2.5 = -2.692582404 Simplifying t + 2.5 = -2.692582404 Reorder the terms: 2.5 + t = -2.692582404 Solving 2.5 + t = -2.692582404 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 = -2.692582404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -2.692582404 + -2.5 t = -2.692582404 + -2.5 Combine like terms: -2.692582404 + -2.5 = -5.192582404 t = -5.192582404 Simplifying t = -5.192582404Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.192582404, -5.192582404}Solution
t = {0, 0.192582404, -5.192582404}
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