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Simplifying 12t(t2 + 3t + -5) = 0 Reorder the terms: 12t(-5 + 3t + t2) = 0 (-5 * 12t + 3t * 12t + t2 * 12t) = 0 (-60t + 36t2 + 12t3) = 0 Solving -60t + 36t2 + 12t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '12t'. 12t(-5 + 3t + 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 '(-5 + 3t + t2)' equal to zero and attempt to solve: Simplifying -5 + 3t + t2 = 0 Solving -5 + 3t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + 3t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + 3t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 3t + t2 = 0 + 5 3t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 3t + t2 = 5 The t term is 3t. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3t + 2.25 + t2 = 5 + 2.25 Reorder the terms: 2.25 + 3t + t2 = 5 + 2.25 Combine like terms: 5 + 2.25 = 7.25 2.25 + 3t + t2 = 7.25 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (t + 1.5) equal to 2.692582404 and -2.692582404.Subproblem 1
t + 1.5 = 2.692582404 Simplifying t + 1.5 = 2.692582404 Reorder the terms: 1.5 + t = 2.692582404 Solving 1.5 + t = 2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 2.692582404 + -1.5 t = 2.692582404 + -1.5 Combine like terms: 2.692582404 + -1.5 = 1.192582404 t = 1.192582404 Simplifying t = 1.192582404Subproblem 2
t + 1.5 = -2.692582404 Simplifying t + 1.5 = -2.692582404 Reorder the terms: 1.5 + t = -2.692582404 Solving 1.5 + t = -2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -2.692582404 + -1.5 t = -2.692582404 + -1.5 Combine like terms: -2.692582404 + -1.5 = -4.192582404 t = -4.192582404 Simplifying t = -4.192582404Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.192582404, -4.192582404}Solution
t = {0, 1.192582404, -4.192582404}
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