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Simplifying (2q + 1)(q + -1) = 18 Reorder the terms: (1 + 2q)(q + -1) = 18 Reorder the terms: (1 + 2q)(-1 + q) = 18 Multiply (1 + 2q) * (-1 + q) (1(-1 + q) + 2q * (-1 + q)) = 18 ((-1 * 1 + q * 1) + 2q * (-1 + q)) = 18 ((-1 + 1q) + 2q * (-1 + q)) = 18 (-1 + 1q + (-1 * 2q + q * 2q)) = 18 (-1 + 1q + (-2q + 2q2)) = 18 Combine like terms: 1q + -2q = -1q (-1 + -1q + 2q2) = 18 Solving -1 + -1q + 2q2 = 18 Solving for variable 'q'. Reorder the terms: -1 + -18 + -1q + 2q2 = 18 + -18 Combine like terms: -1 + -18 = -19 -19 + -1q + 2q2 = 18 + -18 Combine like terms: 18 + -18 = 0 -19 + -1q + 2q2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9.5 + -0.5q + q2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + -0.5q + 9.5 + q2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + -0.5q + q2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + -0.5q + q2 = 0 + 9.5 -0.5q + q2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 -0.5q + q2 = 9.5 The q term is -0.5q. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5q + 0.0625 + q2 = 9.5 + 0.0625 Reorder the terms: 0.0625 + -0.5q + q2 = 9.5 + 0.0625 Combine like terms: 9.5 + 0.0625 = 9.5625 0.0625 + -0.5q + q2 = 9.5625 Factor a perfect square on the left side: (q + -0.25)(q + -0.25) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (q + -0.25) equal to 3.092329219 and -3.092329219.Subproblem 1
q + -0.25 = 3.092329219 Simplifying q + -0.25 = 3.092329219 Reorder the terms: -0.25 + q = 3.092329219 Solving -0.25 + q = 3.092329219 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + q = 3.092329219 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + q = 3.092329219 + 0.25 q = 3.092329219 + 0.25 Combine like terms: 3.092329219 + 0.25 = 3.342329219 q = 3.342329219 Simplifying q = 3.342329219Subproblem 2
q + -0.25 = -3.092329219 Simplifying q + -0.25 = -3.092329219 Reorder the terms: -0.25 + q = -3.092329219 Solving -0.25 + q = -3.092329219 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + q = -3.092329219 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + q = -3.092329219 + 0.25 q = -3.092329219 + 0.25 Combine like terms: -3.092329219 + 0.25 = -2.842329219 q = -2.842329219 Simplifying q = -2.842329219Solution
The solution to the problem is based on the solutions from the subproblems. q = {3.342329219, -2.842329219}
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