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Simplifying (1000 + -1q) * q + -22500 + 400 = 0 Reorder the terms for easier multiplication: q(1000 + -1q) + -22500 + 400 = 0 (1000 * q + -1q * q) + -22500 + 400 = 0 (1000q + -1q2) + -22500 + 400 = 0 Reorder the terms: -22500 + 400 + 1000q + -1q2 = 0 Combine like terms: -22500 + 400 = -22100 -22100 + 1000q + -1q2 = 0 Solving -22100 + 1000q + -1q2 = 0 Solving for variable 'q'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 22100 + -1000q + q2 = 0 Move the constant term to the right: Add '-22100' to each side of the equation. 22100 + -1000q + -22100 + q2 = 0 + -22100 Reorder the terms: 22100 + -22100 + -1000q + q2 = 0 + -22100 Combine like terms: 22100 + -22100 = 0 0 + -1000q + q2 = 0 + -22100 -1000q + q2 = 0 + -22100 Combine like terms: 0 + -22100 = -22100 -1000q + q2 = -22100 The q term is -1000q. Take half its coefficient (-500). Square it (250000) and add it to both sides. Add '250000' to each side of the equation. -1000q + 250000 + q2 = -22100 + 250000 Reorder the terms: 250000 + -1000q + q2 = -22100 + 250000 Combine like terms: -22100 + 250000 = 227900 250000 + -1000q + q2 = 227900 Factor a perfect square on the left side: (q + -500)(q + -500) = 227900 Calculate the square root of the right side: 477.388730491 Break this problem into two subproblems by setting (q + -500) equal to 477.388730491 and -477.388730491.Subproblem 1
q + -500 = 477.388730491 Simplifying q + -500 = 477.388730491 Reorder the terms: -500 + q = 477.388730491 Solving -500 + q = 477.388730491 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + q = 477.388730491 + 500 Combine like terms: -500 + 500 = 0 0 + q = 477.388730491 + 500 q = 477.388730491 + 500 Combine like terms: 477.388730491 + 500 = 977.388730491 q = 977.388730491 Simplifying q = 977.388730491Subproblem 2
q + -500 = -477.388730491 Simplifying q + -500 = -477.388730491 Reorder the terms: -500 + q = -477.388730491 Solving -500 + q = -477.388730491 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + q = -477.388730491 + 500 Combine like terms: -500 + 500 = 0 0 + q = -477.388730491 + 500 q = -477.388730491 + 500 Combine like terms: -477.388730491 + 500 = 22.611269509 q = 22.611269509 Simplifying q = 22.611269509Solution
The solution to the problem is based on the solutions from the subproblems. q = {977.388730491, 22.611269509}
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