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Simplifying 80(Q) = 40 + 8(Q) + 2Q2 Solving 80Q = 40 + 8Q + 2Q2 Solving for variable 'Q'. Reorder the terms: -40 + 80Q + -8Q + -2Q2 = 40 + 8Q + 2Q2 + -40 + -8Q + -2Q2 Combine like terms: 80Q + -8Q = 72Q -40 + 72Q + -2Q2 = 40 + 8Q + 2Q2 + -40 + -8Q + -2Q2 Reorder the terms: -40 + 72Q + -2Q2 = 40 + -40 + 8Q + -8Q + 2Q2 + -2Q2 Combine like terms: 40 + -40 = 0 -40 + 72Q + -2Q2 = 0 + 8Q + -8Q + 2Q2 + -2Q2 -40 + 72Q + -2Q2 = 8Q + -8Q + 2Q2 + -2Q2 Combine like terms: 8Q + -8Q = 0 -40 + 72Q + -2Q2 = 0 + 2Q2 + -2Q2 -40 + 72Q + -2Q2 = 2Q2 + -2Q2 Combine like terms: 2Q2 + -2Q2 = 0 -40 + 72Q + -2Q2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-20 + 36Q + -1Q2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-20 + 36Q + -1Q2)' equal to zero and attempt to solve: Simplifying -20 + 36Q + -1Q2 = 0 Solving -20 + 36Q + -1Q2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 20 + -36Q + Q2 = 0 Move the constant term to the right: Add '-20' to each side of the equation. 20 + -36Q + -20 + Q2 = 0 + -20 Reorder the terms: 20 + -20 + -36Q + Q2 = 0 + -20 Combine like terms: 20 + -20 = 0 0 + -36Q + Q2 = 0 + -20 -36Q + Q2 = 0 + -20 Combine like terms: 0 + -20 = -20 -36Q + Q2 = -20 The Q term is -36Q. Take half its coefficient (-18). Square it (324) and add it to both sides. Add '324' to each side of the equation. -36Q + 324 + Q2 = -20 + 324 Reorder the terms: 324 + -36Q + Q2 = -20 + 324 Combine like terms: -20 + 324 = 304 324 + -36Q + Q2 = 304 Factor a perfect square on the left side: (Q + -18)(Q + -18) = 304 Calculate the square root of the right side: 17.435595774 Break this problem into two subproblems by setting (Q + -18) equal to 17.435595774 and -17.435595774.Subproblem 1
Q + -18 = 17.435595774 Simplifying Q + -18 = 17.435595774 Reorder the terms: -18 + Q = 17.435595774 Solving -18 + Q = 17.435595774 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '18' to each side of the equation. -18 + 18 + Q = 17.435595774 + 18 Combine like terms: -18 + 18 = 0 0 + Q = 17.435595774 + 18 Q = 17.435595774 + 18 Combine like terms: 17.435595774 + 18 = 35.435595774 Q = 35.435595774 Simplifying Q = 35.435595774Subproblem 2
Q + -18 = -17.435595774 Simplifying Q + -18 = -17.435595774 Reorder the terms: -18 + Q = -17.435595774 Solving -18 + Q = -17.435595774 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '18' to each side of the equation. -18 + 18 + Q = -17.435595774 + 18 Combine like terms: -18 + 18 = 0 0 + Q = -17.435595774 + 18 Q = -17.435595774 + 18 Combine like terms: -17.435595774 + 18 = 0.564404226 Q = 0.564404226 Simplifying Q = 0.564404226Solution
The solution to the problem is based on the solutions from the subproblems. Q = {35.435595774, 0.564404226}Solution
Q = {35.435595774, 0.564404226}
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