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Simplifying q(1 + -0.5q) = 0.233 (1 * q + -0.5q * q) = 0.233 (1q + -0.5q2) = 0.233 Solving 1q + -0.5q2 = 0.233 Solving for variable 'q'. Reorder the terms: -0.233 + 1q + -0.5q2 = 0.233 + -0.233 Combine like terms: 0.233 + -0.233 = 0.000 -0.233 + 1q + -0.5q2 = 0.000 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 0.466 + -2q + q2 = 0.00 Move the constant term to the right: Add '-0.466' to each side of the equation. 0.466 + -2q + -0.466 + q2 = 0.00 + -0.466 Reorder the terms: 0.466 + -0.466 + -2q + q2 = 0.00 + -0.466 Combine like terms: 0.466 + -0.466 = 0.000 0.000 + -2q + q2 = 0.00 + -0.466 -2q + q2 = 0.00 + -0.466 Combine like terms: 0.00 + -0.466 = -0.466 -2q + q2 = -0.466 The q term is -2q. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2q + 1 + q2 = -0.466 + 1 Reorder the terms: 1 + -2q + q2 = -0.466 + 1 Combine like terms: -0.466 + 1 = 0.534 1 + -2q + q2 = 0.534 Factor a perfect square on the left side: (q + -1)(q + -1) = 0.534 Calculate the square root of the right side: 0.730753036 Break this problem into two subproblems by setting (q + -1) equal to 0.730753036 and -0.730753036.Subproblem 1
q + -1 = 0.730753036 Simplifying q + -1 = 0.730753036 Reorder the terms: -1 + q = 0.730753036 Solving -1 + q = 0.730753036 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + q = 0.730753036 + 1 Combine like terms: -1 + 1 = 0 0 + q = 0.730753036 + 1 q = 0.730753036 + 1 Combine like terms: 0.730753036 + 1 = 1.730753036 q = 1.730753036 Simplifying q = 1.730753036Subproblem 2
q + -1 = -0.730753036 Simplifying q + -1 = -0.730753036 Reorder the terms: -1 + q = -0.730753036 Solving -1 + q = -0.730753036 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + q = -0.730753036 + 1 Combine like terms: -1 + 1 = 0 0 + q = -0.730753036 + 1 q = -0.730753036 + 1 Combine like terms: -0.730753036 + 1 = 0.269246964 q = 0.269246964 Simplifying q = 0.269246964Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.730753036, 0.269246964}
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