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Simplifying -1t3 + 15t2 + -27t = 0 Reorder the terms: -27t + 15t2 + -1t3 = 0 Solving -27t + 15t2 + -1t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't'. t(-27 + 15t + -1t2) = 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 '(-27 + 15t + -1t2)' equal to zero and attempt to solve: Simplifying -27 + 15t + -1t2 = 0 Solving -27 + 15t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 27 + -15t + t2 = 0 Move the constant term to the right: Add '-27' to each side of the equation. 27 + -15t + -27 + t2 = 0 + -27 Reorder the terms: 27 + -27 + -15t + t2 = 0 + -27 Combine like terms: 27 + -27 = 0 0 + -15t + t2 = 0 + -27 -15t + t2 = 0 + -27 Combine like terms: 0 + -27 = -27 -15t + t2 = -27 The t term is -15t. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15t + 56.25 + t2 = -27 + 56.25 Reorder the terms: 56.25 + -15t + t2 = -27 + 56.25 Combine like terms: -27 + 56.25 = 29.25 56.25 + -15t + t2 = 29.25 Factor a perfect square on the left side: (t + -7.5)(t + -7.5) = 29.25 Calculate the square root of the right side: 5.408326913 Break this problem into two subproblems by setting (t + -7.5) equal to 5.408326913 and -5.408326913.Subproblem 1
t + -7.5 = 5.408326913 Simplifying t + -7.5 = 5.408326913 Reorder the terms: -7.5 + t = 5.408326913 Solving -7.5 + t = 5.408326913 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = 5.408326913 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = 5.408326913 + 7.5 t = 5.408326913 + 7.5 Combine like terms: 5.408326913 + 7.5 = 12.908326913 t = 12.908326913 Simplifying t = 12.908326913Subproblem 2
t + -7.5 = -5.408326913 Simplifying t + -7.5 = -5.408326913 Reorder the terms: -7.5 + t = -5.408326913 Solving -7.5 + t = -5.408326913 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = -5.408326913 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = -5.408326913 + 7.5 t = -5.408326913 + 7.5 Combine like terms: -5.408326913 + 7.5 = 2.091673087 t = 2.091673087 Simplifying t = 2.091673087Solution
The solution to the problem is based on the solutions from the subproblems. t = {12.908326913, 2.091673087}Solution
t = {0, 12.908326913, 2.091673087}
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