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Simplifying 12t3 + 12t2 + -6t = 0 Reorder the terms: -6t + 12t2 + 12t3 = 0 Solving -6t + 12t2 + 12t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '6t'. 6t(-1 + 2t + 2t2) = 0 Ignore the factor 6.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 + 2t + 2t2)' equal to zero and attempt to solve: Simplifying -1 + 2t + 2t2 = 0 Solving -1 + 2t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t + t2 = 0 + 0.5 t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 t + t2 = 0.5 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + t2 = 0.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 1.25 + t2 = 0.75 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (t + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
t + 0.5 = 0.866025404 Simplifying t + 0.5 = 0.866025404 Reorder the terms: 0.5 + t = 0.866025404 Solving 0.5 + t = 0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 0.866025404 + -0.5 t = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 t = 0.366025404 Simplifying t = 0.366025404Subproblem 2
t + 0.5 = -0.866025404 Simplifying t + 0.5 = -0.866025404 Reorder the terms: 0.5 + t = -0.866025404 Solving 0.5 + t = -0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -0.866025404 + -0.5 t = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 t = -1.366025404 Simplifying t = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.366025404, -1.366025404}Solution
t = {0, 0.366025404, -1.366025404}
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