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Simplifying 2(6 + -1q) * q = 1 Reorder the terms for easier multiplication: 2q(6 + -1q) = 1 (6 * 2q + -1q * 2q) = 1 (12q + -2q2) = 1 Solving 12q + -2q2 = 1 Solving for variable 'q'. Reorder the terms: -1 + 12q + -2q2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 12q + -2q2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 0.5 + -6q + q2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -6q + -0.5 + q2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -6q + q2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -6q + q2 = 0 + -0.5 -6q + q2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -6q + q2 = -0.5 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 = -0.5 + 9 Reorder the terms: 9 + -6q + q2 = -0.5 + 9 Combine like terms: -0.5 + 9 = 8.5 9 + -6q + q2 = 8.5 Factor a perfect square on the left side: (q + -3)(q + -3) = 8.5 Calculate the square root of the right side: 2.915475947 Break this problem into two subproblems by setting (q + -3) equal to 2.915475947 and -2.915475947.Subproblem 1
q + -3 = 2.915475947 Simplifying q + -3 = 2.915475947 Reorder the terms: -3 + q = 2.915475947 Solving -3 + q = 2.915475947 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 = 2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + q = 2.915475947 + 3 q = 2.915475947 + 3 Combine like terms: 2.915475947 + 3 = 5.915475947 q = 5.915475947 Simplifying q = 5.915475947Subproblem 2
q + -3 = -2.915475947 Simplifying q + -3 = -2.915475947 Reorder the terms: -3 + q = -2.915475947 Solving -3 + q = -2.915475947 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 = -2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + q = -2.915475947 + 3 q = -2.915475947 + 3 Combine like terms: -2.915475947 + 3 = 0.084524053 q = 0.084524053 Simplifying q = 0.084524053Solution
The solution to the problem is based on the solutions from the subproblems. q = {5.915475947, 0.084524053}
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