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Simplifying 4q2 + 2q = 3q2 + -4q + 18 Reorder the terms: 2q + 4q2 = 3q2 + -4q + 18 Reorder the terms: 2q + 4q2 = 18 + -4q + 3q2 Solving 2q + 4q2 = 18 + -4q + 3q2 Solving for variable 'q'. Reorder the terms: -18 + 2q + 4q + 4q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 2q + 4q = 6q -18 + 6q + 4q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 4q2 + -3q2 = 1q2 -18 + 6q + 1q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Reorder the terms: -18 + 6q + 1q2 = 18 + -18 + -4q + 4q + 3q2 + -3q2 Combine like terms: 18 + -18 = 0 -18 + 6q + 1q2 = 0 + -4q + 4q + 3q2 + -3q2 -18 + 6q + 1q2 = -4q + 4q + 3q2 + -3q2 Combine like terms: -4q + 4q = 0 -18 + 6q + 1q2 = 0 + 3q2 + -3q2 -18 + 6q + 1q2 = 3q2 + -3q2 Combine like terms: 3q2 + -3q2 = 0 -18 + 6q + 1q2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 6q + 18 + q2 = 0 + 18 Reorder the terms: -18 + 18 + 6q + q2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 6q + q2 = 0 + 18 6q + q2 = 0 + 18 Combine like terms: 0 + 18 = 18 6q + q2 = 18 The q term is 6q. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6q + 9 + q2 = 18 + 9 Reorder the terms: 9 + 6q + q2 = 18 + 9 Combine like terms: 18 + 9 = 27 9 + 6q + q2 = 27 Factor a perfect square on the left side: (q + 3)(q + 3) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (q + 3) equal to 5.196152423 and -5.196152423.Subproblem 1
q + 3 = 5.196152423 Simplifying q + 3 = 5.196152423 Reorder the terms: 3 + q = 5.196152423 Solving 3 + q = 5.196152423 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + q = 5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + q = 5.196152423 + -3 q = 5.196152423 + -3 Combine like terms: 5.196152423 + -3 = 2.196152423 q = 2.196152423 Simplifying q = 2.196152423Subproblem 2
q + 3 = -5.196152423 Simplifying q + 3 = -5.196152423 Reorder the terms: 3 + q = -5.196152423 Solving 3 + q = -5.196152423 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + q = -5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + q = -5.196152423 + -3 q = -5.196152423 + -3 Combine like terms: -5.196152423 + -3 = -8.196152423 q = -8.196152423 Simplifying q = -8.196152423Solution
The solution to the problem is based on the solutions from the subproblems. q = {2.196152423, -8.196152423}
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