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Simplifying t = 4t3 + -36t2 + 80t Reorder the terms: t = 80t + -36t2 + 4t3 Solving t = 80t + -36t2 + 4t3 Solving for variable 't'. Combine like terms: t + -80t = -79t -79t + 36t2 + -4t3 = 80t + -36t2 + 4t3 + -80t + 36t2 + -4t3 Reorder the terms: -79t + 36t2 + -4t3 = 80t + -80t + -36t2 + 36t2 + 4t3 + -4t3 Combine like terms: 80t + -80t = 0 -79t + 36t2 + -4t3 = 0 + -36t2 + 36t2 + 4t3 + -4t3 -79t + 36t2 + -4t3 = -36t2 + 36t2 + 4t3 + -4t3 Combine like terms: -36t2 + 36t2 = 0 -79t + 36t2 + -4t3 = 0 + 4t3 + -4t3 -79t + 36t2 + -4t3 = 4t3 + -4t3 Combine like terms: 4t3 + -4t3 = 0 -79t + 36t2 + -4t3 = 0 Factor out the Greatest Common Factor (GCF), 't'. t(-79 + 36t + -4t2) = 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 '(-79 + 36t + -4t2)' equal to zero and attempt to solve: Simplifying -79 + 36t + -4t2 = 0 Solving -79 + 36t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 19.75 + -9t + t2 = 0 Move the constant term to the right: Add '-19.75' to each side of the equation. 19.75 + -9t + -19.75 + t2 = 0 + -19.75 Reorder the terms: 19.75 + -19.75 + -9t + t2 = 0 + -19.75 Combine like terms: 19.75 + -19.75 = 0.00 0.00 + -9t + t2 = 0 + -19.75 -9t + t2 = 0 + -19.75 Combine like terms: 0 + -19.75 = -19.75 -9t + t2 = -19.75 The t term is -9t. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t + 20.25 + t2 = -19.75 + 20.25 Reorder the terms: 20.25 + -9t + t2 = -19.75 + 20.25 Combine like terms: -19.75 + 20.25 = 0.5 20.25 + -9t + t2 = 0.5 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (t + -4.5) equal to 0.707106781 and -0.707106781.Subproblem 1
t + -4.5 = 0.707106781 Simplifying t + -4.5 = 0.707106781 Reorder the terms: -4.5 + t = 0.707106781 Solving -4.5 + t = 0.707106781 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 0.707106781 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 0.707106781 + 4.5 t = 0.707106781 + 4.5 Combine like terms: 0.707106781 + 4.5 = 5.207106781 t = 5.207106781 Simplifying t = 5.207106781Subproblem 2
t + -4.5 = -0.707106781 Simplifying t + -4.5 = -0.707106781 Reorder the terms: -4.5 + t = -0.707106781 Solving -4.5 + t = -0.707106781 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -0.707106781 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -0.707106781 + 4.5 t = -0.707106781 + 4.5 Combine like terms: -0.707106781 + 4.5 = 3.792893219 t = 3.792893219 Simplifying t = 3.792893219Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.207106781, 3.792893219}Solution
t = {0, 5.207106781, 3.792893219}
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