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Simplifying 0 = 80 + -20Q + 0.18Q2 Solving 0 = 80 + -20Q + 0.18Q2 Solving for variable 'Q'. Combine like terms: 0 + -80 = -80 -80 + 20Q + -0.18Q2 = 80 + -20Q + 0.18Q2 + -80 + 20Q + -0.18Q2 Reorder the terms: -80 + 20Q + -0.18Q2 = 80 + -80 + -20Q + 20Q + 0.18Q2 + -0.18Q2 Combine like terms: 80 + -80 = 0 -80 + 20Q + -0.18Q2 = 0 + -20Q + 20Q + 0.18Q2 + -0.18Q2 -80 + 20Q + -0.18Q2 = -20Q + 20Q + 0.18Q2 + -0.18Q2 Combine like terms: -20Q + 20Q = 0 -80 + 20Q + -0.18Q2 = 0 + 0.18Q2 + -0.18Q2 -80 + 20Q + -0.18Q2 = 0.18Q2 + -0.18Q2 Combine like terms: 0.18Q2 + -0.18Q2 = 0.00 -80 + 20Q + -0.18Q2 = 0.00 Begin completing the square. Divide all terms by -0.18 the coefficient of the squared term: Divide each side by '-0.18'. 444.4444444 + -111.1111111Q + Q2 = 0 Move the constant term to the right: Add '-444.4444444' to each side of the equation. 444.4444444 + -111.1111111Q + -444.4444444 + Q2 = 0 + -444.4444444 Reorder the terms: 444.4444444 + -444.4444444 + -111.1111111Q + Q2 = 0 + -444.4444444 Combine like terms: 444.4444444 + -444.4444444 = 0.0000000 0.0000000 + -111.1111111Q + Q2 = 0 + -444.4444444 -111.1111111Q + Q2 = 0 + -444.4444444 Combine like terms: 0 + -444.4444444 = -444.4444444 -111.1111111Q + Q2 = -444.4444444 The Q term is -111.1111111Q. Take half its coefficient (-55.55555555). Square it (3086.419752) and add it to both sides. Add '3086.419752' to each side of the equation. -111.1111111Q + 3086.419752 + Q2 = -444.4444444 + 3086.419752 Reorder the terms: 3086.419752 + -111.1111111Q + Q2 = -444.4444444 + 3086.419752 Combine like terms: -444.4444444 + 3086.419752 = 2641.9753076 3086.419752 + -111.1111111Q + Q2 = 2641.9753076 Factor a perfect square on the left side: (Q + -55.55555555)(Q + -55.55555555) = 2641.9753076 Calculate the square root of the right side: 51.400148906 Break this problem into two subproblems by setting (Q + -55.55555555) equal to 51.400148906 and -51.400148906.Subproblem 1
Q + -55.55555555 = 51.400148906 Simplifying Q + -55.55555555 = 51.400148906 Reorder the terms: -55.55555555 + Q = 51.400148906 Solving -55.55555555 + Q = 51.400148906 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '55.55555555' to each side of the equation. -55.55555555 + 55.55555555 + Q = 51.400148906 + 55.55555555 Combine like terms: -55.55555555 + 55.55555555 = 0.00000000 0.00000000 + Q = 51.400148906 + 55.55555555 Q = 51.400148906 + 55.55555555 Combine like terms: 51.400148906 + 55.55555555 = 106.955704456 Q = 106.955704456 Simplifying Q = 106.955704456Subproblem 2
Q + -55.55555555 = -51.400148906 Simplifying Q + -55.55555555 = -51.400148906 Reorder the terms: -55.55555555 + Q = -51.400148906 Solving -55.55555555 + Q = -51.400148906 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '55.55555555' to each side of the equation. -55.55555555 + 55.55555555 + Q = -51.400148906 + 55.55555555 Combine like terms: -55.55555555 + 55.55555555 = 0.00000000 0.00000000 + Q = -51.400148906 + 55.55555555 Q = -51.400148906 + 55.55555555 Combine like terms: -51.400148906 + 55.55555555 = 4.155406644 Q = 4.155406644 Simplifying Q = 4.155406644Solution
The solution to the problem is based on the solutions from the subproblems. Q = {106.955704456, 4.155406644}
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