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Simplifying 6t3 + 12t2 + -4t = 0 Reorder the terms: -4t + 12t2 + 6t3 = 0 Solving -4t + 12t2 + 6t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2t'. 2t(-2 + 6t + 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 '(-2 + 6t + 3t2)' equal to zero and attempt to solve: Simplifying -2 + 6t + 3t2 = 0 Solving -2 + 6t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.6666666667 + 2t + t2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + 2t + 0.6666666667 + t2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + 2t + t2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + 2t + t2 = 0 + 0.6666666667 2t + t2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 2t + t2 = 0.6666666667 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 0.6666666667 + 1 Reorder the terms: 1 + 2t + t2 = 0.6666666667 + 1 Combine like terms: 0.6666666667 + 1 = 1.6666666667 1 + 2t + t2 = 1.6666666667 Factor a perfect square on the left side: (t + 1)(t + 1) = 1.6666666667 Calculate the square root of the right side: 1.290994449 Break this problem into two subproblems by setting (t + 1) equal to 1.290994449 and -1.290994449.Subproblem 1
t + 1 = 1.290994449 Simplifying t + 1 = 1.290994449 Reorder the terms: 1 + t = 1.290994449 Solving 1 + t = 1.290994449 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 1.290994449 + -1 Combine like terms: 1 + -1 = 0 0 + t = 1.290994449 + -1 t = 1.290994449 + -1 Combine like terms: 1.290994449 + -1 = 0.290994449 t = 0.290994449 Simplifying t = 0.290994449Subproblem 2
t + 1 = -1.290994449 Simplifying t + 1 = -1.290994449 Reorder the terms: 1 + t = -1.290994449 Solving 1 + t = -1.290994449 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -1.290994449 + -1 Combine like terms: 1 + -1 = 0 0 + t = -1.290994449 + -1 t = -1.290994449 + -1 Combine like terms: -1.290994449 + -1 = -2.290994449 t = -2.290994449 Simplifying t = -2.290994449Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.290994449, -2.290994449}Solution
t = {0, 0.290994449, -2.290994449}
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