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Simplifying 56 + -0.02Q = 20 + -0.14Q + 0.0003Q2 Solving 56 + -0.02Q = 20 + -0.14Q + 0.0003Q2 Solving for variable 'Q'. Reorder the terms: 56 + -20 + -0.02Q + 0.14Q + -0.0003Q2 = 20 + -0.14Q + 0.0003Q2 + -20 + 0.14Q + -0.0003Q2 Combine like terms: 56 + -20 = 36 36 + -0.02Q + 0.14Q + -0.0003Q2 = 20 + -0.14Q + 0.0003Q2 + -20 + 0.14Q + -0.0003Q2 Combine like terms: -0.02Q + 0.14Q = 0.12Q 36 + 0.12Q + -0.0003Q2 = 20 + -0.14Q + 0.0003Q2 + -20 + 0.14Q + -0.0003Q2 Reorder the terms: 36 + 0.12Q + -0.0003Q2 = 20 + -20 + -0.14Q + 0.14Q + 0.0003Q2 + -0.0003Q2 Combine like terms: 20 + -20 = 0 36 + 0.12Q + -0.0003Q2 = 0 + -0.14Q + 0.14Q + 0.0003Q2 + -0.0003Q2 36 + 0.12Q + -0.0003Q2 = -0.14Q + 0.14Q + 0.0003Q2 + -0.0003Q2 Combine like terms: -0.14Q + 0.14Q = 0.00 36 + 0.12Q + -0.0003Q2 = 0.00 + 0.0003Q2 + -0.0003Q2 36 + 0.12Q + -0.0003Q2 = 0.0003Q2 + -0.0003Q2 Combine like terms: 0.0003Q2 + -0.0003Q2 = 0.0000 36 + 0.12Q + -0.0003Q2 = 0.0000 Begin completing the square. Divide all terms by -0.0003 the coefficient of the squared term: Divide each side by '-0.0003'. -120000 + -400Q + Q2 = 0 Move the constant term to the right: Add '120000' to each side of the equation. -120000 + -400Q + 120000 + Q2 = 0 + 120000 Reorder the terms: -120000 + 120000 + -400Q + Q2 = 0 + 120000 Combine like terms: -120000 + 120000 = 0 0 + -400Q + Q2 = 0 + 120000 -400Q + Q2 = 0 + 120000 Combine like terms: 0 + 120000 = 120000 -400Q + Q2 = 120000 The Q term is -400Q. Take half its coefficient (-200). Square it (40000) and add it to both sides. Add '40000' to each side of the equation. -400Q + 40000 + Q2 = 120000 + 40000 Reorder the terms: 40000 + -400Q + Q2 = 120000 + 40000 Combine like terms: 120000 + 40000 = 160000 40000 + -400Q + Q2 = 160000 Factor a perfect square on the left side: (Q + -200)(Q + -200) = 160000 Calculate the square root of the right side: 400 Break this problem into two subproblems by setting (Q + -200) equal to 400 and -400.Subproblem 1
Q + -200 = 400 Simplifying Q + -200 = 400 Reorder the terms: -200 + Q = 400 Solving -200 + Q = 400 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + Q = 400 + 200 Combine like terms: -200 + 200 = 0 0 + Q = 400 + 200 Q = 400 + 200 Combine like terms: 400 + 200 = 600 Q = 600 Simplifying Q = 600Subproblem 2
Q + -200 = -400 Simplifying Q + -200 = -400 Reorder the terms: -200 + Q = -400 Solving -200 + Q = -400 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + Q = -400 + 200 Combine like terms: -200 + 200 = 0 0 + Q = -400 + 200 Q = -400 + 200 Combine like terms: -400 + 200 = -200 Q = -200 Simplifying Q = -200Solution
The solution to the problem is based on the solutions from the subproblems. Q = {600, -200}
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