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Simplifying 3q(3q) + 16q = -5 Remove parenthesis around (3q) 3q * 3q + 16q = -5 Reorder the terms for easier multiplication: 3 * 3q * q + 16q = -5 Multiply 3 * 3 9q * q + 16q = -5 Multiply q * q 9q2 + 16q = -5 Reorder the terms: 16q + 9q2 = -5 Solving 16q + 9q2 = -5 Solving for variable 'q'. Reorder the terms: 5 + 16q + 9q2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + 16q + 9q2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.5555555556 + 1.777777778q + q2 = 0 Move the constant term to the right: Add '-0.5555555556' to each side of the equation. 0.5555555556 + 1.777777778q + -0.5555555556 + q2 = 0 + -0.5555555556 Reorder the terms: 0.5555555556 + -0.5555555556 + 1.777777778q + q2 = 0 + -0.5555555556 Combine like terms: 0.5555555556 + -0.5555555556 = 0.0000000000 0.0000000000 + 1.777777778q + q2 = 0 + -0.5555555556 1.777777778q + q2 = 0 + -0.5555555556 Combine like terms: 0 + -0.5555555556 = -0.5555555556 1.777777778q + q2 = -0.5555555556 The q term is 1.777777778q. Take half its coefficient (0.888888889). Square it (0.7901234570) and add it to both sides. Add '0.7901234570' to each side of the equation. 1.777777778q + 0.7901234570 + q2 = -0.5555555556 + 0.7901234570 Reorder the terms: 0.7901234570 + 1.777777778q + q2 = -0.5555555556 + 0.7901234570 Combine like terms: -0.5555555556 + 0.7901234570 = 0.2345679014 0.7901234570 + 1.777777778q + q2 = 0.2345679014 Factor a perfect square on the left side: (q + 0.888888889)(q + 0.888888889) = 0.2345679014 Calculate the square root of the right side: 0.484322105 Break this problem into two subproblems by setting (q + 0.888888889) equal to 0.484322105 and -0.484322105.Subproblem 1
q + 0.888888889 = 0.484322105 Simplifying q + 0.888888889 = 0.484322105 Reorder the terms: 0.888888889 + q = 0.484322105 Solving 0.888888889 + q = 0.484322105 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + q = 0.484322105 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + q = 0.484322105 + -0.888888889 q = 0.484322105 + -0.888888889 Combine like terms: 0.484322105 + -0.888888889 = -0.404566784 q = -0.404566784 Simplifying q = -0.404566784Subproblem 2
q + 0.888888889 = -0.484322105 Simplifying q + 0.888888889 = -0.484322105 Reorder the terms: 0.888888889 + q = -0.484322105 Solving 0.888888889 + q = -0.484322105 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + q = -0.484322105 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + q = -0.484322105 + -0.888888889 q = -0.484322105 + -0.888888889 Combine like terms: -0.484322105 + -0.888888889 = -1.373210994 q = -1.373210994 Simplifying q = -1.373210994Solution
The solution to the problem is based on the solutions from the subproblems. q = {-0.404566784, -1.373210994}
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