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Simplifying -27t2 + 27t + 6 = 0 Reorder the terms: 6 + 27t + -27t2 = 0 Solving 6 + 27t + -27t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '3'. 3(2 + 9t + -9t2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(2 + 9t + -9t2)' equal to zero and attempt to solve: Simplifying 2 + 9t + -9t2 = 0 Solving 2 + 9t + -9t2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.2222222222 + -1t + t2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + -1t + 0.2222222222 + t2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + -1t + t2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + -1t + t2 = 0 + 0.2222222222 -1t + t2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 -1t + t2 = 0.2222222222 The t term is -1t. 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. -1t + 0.25 + t2 = 0.2222222222 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 0.2222222222 + 0.25 Combine like terms: 0.2222222222 + 0.25 = 0.4722222222 0.25 + -1t + t2 = 0.4722222222 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 0.4722222222 Calculate the square root of the right side: 0.687184271 Break this problem into two subproblems by setting (t + -0.5) equal to 0.687184271 and -0.687184271.Subproblem 1
t + -0.5 = 0.687184271 Simplifying t + -0.5 = 0.687184271 Reorder the terms: -0.5 + t = 0.687184271 Solving -0.5 + t = 0.687184271 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 = 0.687184271 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 0.687184271 + 0.5 t = 0.687184271 + 0.5 Combine like terms: 0.687184271 + 0.5 = 1.187184271 t = 1.187184271 Simplifying t = 1.187184271Subproblem 2
t + -0.5 = -0.687184271 Simplifying t + -0.5 = -0.687184271 Reorder the terms: -0.5 + t = -0.687184271 Solving -0.5 + t = -0.687184271 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 = -0.687184271 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -0.687184271 + 0.5 t = -0.687184271 + 0.5 Combine like terms: -0.687184271 + 0.5 = -0.187184271 t = -0.187184271 Simplifying t = -0.187184271Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.187184271, -0.187184271}Solution
t = {1.187184271, -0.187184271}
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