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Simplifying -6t3 + -4t2 + 12t = 0 Reorder the terms: 12t + -4t2 + -6t3 = 0 Solving 12t + -4t2 + -6t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2t'. 2t(6 + -2t + -3t2) = 0 Ignore the factor 2.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 '(6 + -2t + -3t2)' equal to zero and attempt to solve: Simplifying 6 + -2t + -3t2 = 0 Solving 6 + -2t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -2 + 0.6666666667t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.6666666667t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + 0.6666666667t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.6666666667t + t2 = 0 + 2 0.6666666667t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.6666666667t + t2 = 2 The t term is 0.6666666667t. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667t + 0.1111111112 + t2 = 2 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667t + t2 = 2 + 0.1111111112 Combine like terms: 2 + 0.1111111112 = 2.1111111112 0.1111111112 + 0.6666666667t + t2 = 2.1111111112 Factor a perfect square on the left side: (t + 0.3333333334)(t + 0.3333333334) = 2.1111111112 Calculate the square root of the right side: 1.452966315 Break this problem into two subproblems by setting (t + 0.3333333334) equal to 1.452966315 and -1.452966315.Subproblem 1
t + 0.3333333334 = 1.452966315 Simplifying t + 0.3333333334 = 1.452966315 Reorder the terms: 0.3333333334 + t = 1.452966315 Solving 0.3333333334 + t = 1.452966315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + t = 1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + t = 1.452966315 + -0.3333333334 t = 1.452966315 + -0.3333333334 Combine like terms: 1.452966315 + -0.3333333334 = 1.1196329816 t = 1.1196329816 Simplifying t = 1.1196329816Subproblem 2
t + 0.3333333334 = -1.452966315 Simplifying t + 0.3333333334 = -1.452966315 Reorder the terms: 0.3333333334 + t = -1.452966315 Solving 0.3333333334 + t = -1.452966315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + t = -1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + t = -1.452966315 + -0.3333333334 t = -1.452966315 + -0.3333333334 Combine like terms: -1.452966315 + -0.3333333334 = -1.7862996484 t = -1.7862996484 Simplifying t = -1.7862996484Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.1196329816, -1.7862996484}Solution
t = {0, 1.1196329816, -1.7862996484}
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