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Simplifying 18q3 + 6q2 + -6q = 0 Reorder the terms: -6q + 6q2 + 18q3 = 0 Solving -6q + 6q2 + 18q3 = 0 Solving for variable 'q'. Factor out the Greatest Common Factor (GCF), '6q'. 6q(-1 + q + 3q2) = 0 Ignore the factor 6.Subproblem 1
Set the factor 'q' equal to zero and attempt to solve: Simplifying q = 0 Solving q = 0 Move all terms containing q to the left, all other terms to the right. Simplifying q = 0Subproblem 2
Set the factor '(-1 + q + 3q2)' equal to zero and attempt to solve: Simplifying -1 + q + 3q2 = 0 Solving -1 + q + 3q2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + 0.3333333333q + q2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 0.3333333333q + 0.3333333333 + q2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 0.3333333333q + q2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 0.3333333333q + q2 = 0 + 0.3333333333 0.3333333333q + q2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 0.3333333333q + q2 = 0.3333333333 The q term is q. 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. 0.3333333333q + 0.25 + q2 = 0.3333333333 + 0.25 Reorder the terms: 0.25 + 0.3333333333q + q2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 0.25 + 0.3333333333q + q2 = 0.5833333333 Factor a perfect square on the left side: (q + 0.5)(q + 0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (q + 0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
q + 0.5 = 0.763762616 Simplifying q + 0.5 = 0.763762616 Reorder the terms: 0.5 + q = 0.763762616 Solving 0.5 + q = 0.763762616 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + q = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + q = 0.763762616 + -0.5 q = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 q = 0.263762616 Simplifying q = 0.263762616Subproblem 2
q + 0.5 = -0.763762616 Simplifying q + 0.5 = -0.763762616 Reorder the terms: 0.5 + q = -0.763762616 Solving 0.5 + q = -0.763762616 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + q = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + q = -0.763762616 + -0.5 q = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 q = -1.263762616 Simplifying q = -1.263762616Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.263762616, -1.263762616}Solution
q = {0, 0.263762616, -1.263762616}
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