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Simplifying 6t2 + -54t + 20 = 0 Reorder the terms: 20 + -54t + 6t2 = 0 Solving 20 + -54t + 6t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(10 + -27t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(10 + -27t + 3t2)' equal to zero and attempt to solve: Simplifying 10 + -27t + 3t2 = 0 Solving 10 + -27t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.333333333 + -9t + t2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + -9t + -3.333333333 + t2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + -9t + t2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + -9t + t2 = 0 + -3.333333333 -9t + t2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 -9t + t2 = -3.333333333 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 = -3.333333333 + 20.25 Reorder the terms: 20.25 + -9t + t2 = -3.333333333 + 20.25 Combine like terms: -3.333333333 + 20.25 = 16.916666667 20.25 + -9t + t2 = 16.916666667 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 16.916666667 Calculate the square root of the right side: 4.11298756 Break this problem into two subproblems by setting (t + -4.5) equal to 4.11298756 and -4.11298756.Subproblem 1
t + -4.5 = 4.11298756 Simplifying t + -4.5 = 4.11298756 Reorder the terms: -4.5 + t = 4.11298756 Solving -4.5 + t = 4.11298756 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 = 4.11298756 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 4.11298756 + 4.5 t = 4.11298756 + 4.5 Combine like terms: 4.11298756 + 4.5 = 8.61298756 t = 8.61298756 Simplifying t = 8.61298756Subproblem 2
t + -4.5 = -4.11298756 Simplifying t + -4.5 = -4.11298756 Reorder the terms: -4.5 + t = -4.11298756 Solving -4.5 + t = -4.11298756 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 = -4.11298756 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -4.11298756 + 4.5 t = -4.11298756 + 4.5 Combine like terms: -4.11298756 + 4.5 = 0.38701244 t = 0.38701244 Simplifying t = 0.38701244Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.61298756, 0.38701244}Solution
t = {8.61298756, 0.38701244}
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