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Simplifying 12t3 + 12t2 + -36t = 0 Reorder the terms: -36t + 12t2 + 12t3 = 0 Solving -36t + 12t2 + 12t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '12t'. 12t(-3 + t + 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 '(-3 + t + t2)' equal to zero and attempt to solve: Simplifying -3 + t + t2 = 0 Solving -3 + t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + t + 3 + t2 = 0 + 3 Reorder the terms: -3 + 3 + t + t2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + t + t2 = 0 + 3 t + t2 = 0 + 3 Combine like terms: 0 + 3 = 3 t + t2 = 3 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. t + 0.25 + t2 = 3 + 0.25 Reorder the terms: 0.25 + t + t2 = 3 + 0.25 Combine like terms: 3 + 0.25 = 3.25 0.25 + t + t2 = 3.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (t + 0.5) equal to 1.802775638 and -1.802775638.Subproblem 1
t + 0.5 = 1.802775638 Simplifying t + 0.5 = 1.802775638 Reorder the terms: 0.5 + t = 1.802775638 Solving 0.5 + t = 1.802775638 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 = 1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.802775638 + -0.5 t = 1.802775638 + -0.5 Combine like terms: 1.802775638 + -0.5 = 1.302775638 t = 1.302775638 Simplifying t = 1.302775638Subproblem 2
t + 0.5 = -1.802775638 Simplifying t + 0.5 = -1.802775638 Reorder the terms: 0.5 + t = -1.802775638 Solving 0.5 + t = -1.802775638 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 = -1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.802775638 + -0.5 t = -1.802775638 + -0.5 Combine like terms: -1.802775638 + -0.5 = -2.302775638 t = -2.302775638 Simplifying t = -2.302775638Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.302775638, -2.302775638}Solution
t = {0, 1.302775638, -2.302775638}
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