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Simplifying (4q + 2q3 + -1) + (1 + -2q + 6q2) = 0 Reorder the terms: (-1 + 4q + 2q3) + (1 + -2q + 6q2) = 0 Remove parenthesis around (-1 + 4q + 2q3) -1 + 4q + 2q3 + (1 + -2q + 6q2) = 0 Remove parenthesis around (1 + -2q + 6q2) -1 + 4q + 2q3 + 1 + -2q + 6q2 = 0 Reorder the terms: -1 + 1 + 4q + -2q + 6q2 + 2q3 = 0 Combine like terms: -1 + 1 = 0 0 + 4q + -2q + 6q2 + 2q3 = 0 4q + -2q + 6q2 + 2q3 = 0 Combine like terms: 4q + -2q = 2q 2q + 6q2 + 2q3 = 0 Solving 2q + 6q2 + 2q3 = 0 Solving for variable 'q'. Factor out the Greatest Common Factor (GCF), '2q'. 2q(1 + 3q + q2) = 0 Ignore the factor 2.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 + 3q + q2)' equal to zero and attempt to solve: Simplifying 1 + 3q + q2 = 0 Solving 1 + 3q + q2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 3q + -1 + q2 = 0 + -1 Reorder the terms: 1 + -1 + 3q + q2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 3q + q2 = 0 + -1 3q + q2 = 0 + -1 Combine like terms: 0 + -1 = -1 3q + q2 = -1 The q term is 3q. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3q + 2.25 + q2 = -1 + 2.25 Reorder the terms: 2.25 + 3q + q2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + 3q + q2 = 1.25 Factor a perfect square on the left side: (q + 1.5)(q + 1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (q + 1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
q + 1.5 = 1.118033989 Simplifying q + 1.5 = 1.118033989 Reorder the terms: 1.5 + q = 1.118033989 Solving 1.5 + q = 1.118033989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = 1.118033989 + -1.5 q = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 q = -0.381966011 Simplifying q = -0.381966011Subproblem 2
q + 1.5 = -1.118033989 Simplifying q + 1.5 = -1.118033989 Reorder the terms: 1.5 + q = -1.118033989 Solving 1.5 + q = -1.118033989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = -1.118033989 + -1.5 q = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 q = -2.618033989 Simplifying q = -2.618033989Solution
The solution to the problem is based on the solutions from the subproblems. q = {-0.381966011, -2.618033989}Solution
q = {0, -0.381966011, -2.618033989}
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