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Simplifying t3 + -18t2 + -81t = 0 Reorder the terms: -81t + -18t2 + t3 = 0 Solving -81t + -18t2 + t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't'. t(-81 + -18t + t2) = 0Subproblem 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 '(-81 + -18t + t2)' equal to zero and attempt to solve: Simplifying -81 + -18t + t2 = 0 Solving -81 + -18t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '81' to each side of the equation. -81 + -18t + 81 + t2 = 0 + 81 Reorder the terms: -81 + 81 + -18t + t2 = 0 + 81 Combine like terms: -81 + 81 = 0 0 + -18t + t2 = 0 + 81 -18t + t2 = 0 + 81 Combine like terms: 0 + 81 = 81 -18t + t2 = 81 The t term is -18t. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18t + 81 + t2 = 81 + 81 Reorder the terms: 81 + -18t + t2 = 81 + 81 Combine like terms: 81 + 81 = 162 81 + -18t + t2 = 162 Factor a perfect square on the left side: (t + -9)(t + -9) = 162 Calculate the square root of the right side: 12.727922061 Break this problem into two subproblems by setting (t + -9) equal to 12.727922061 and -12.727922061.Subproblem 1
t + -9 = 12.727922061 Simplifying t + -9 = 12.727922061 Reorder the terms: -9 + t = 12.727922061 Solving -9 + t = 12.727922061 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + t = 12.727922061 + 9 Combine like terms: -9 + 9 = 0 0 + t = 12.727922061 + 9 t = 12.727922061 + 9 Combine like terms: 12.727922061 + 9 = 21.727922061 t = 21.727922061 Simplifying t = 21.727922061Subproblem 2
t + -9 = -12.727922061 Simplifying t + -9 = -12.727922061 Reorder the terms: -9 + t = -12.727922061 Solving -9 + t = -12.727922061 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + t = -12.727922061 + 9 Combine like terms: -9 + 9 = 0 0 + t = -12.727922061 + 9 t = -12.727922061 + 9 Combine like terms: -12.727922061 + 9 = -3.727922061 t = -3.727922061 Simplifying t = -3.727922061Solution
The solution to the problem is based on the solutions from the subproblems. t = {21.727922061, -3.727922061}Solution
t = {0, 21.727922061, -3.727922061}
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