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Simplifying 180 = 20Q + 6Q2 Solving 180 = 20Q + 6Q2 Solving for variable 'Q'. Reorder the terms: 180 + -20Q + -6Q2 = 20Q + -20Q + 6Q2 + -6Q2 Combine like terms: 20Q + -20Q = 0 180 + -20Q + -6Q2 = 0 + 6Q2 + -6Q2 180 + -20Q + -6Q2 = 6Q2 + -6Q2 Combine like terms: 6Q2 + -6Q2 = 0 180 + -20Q + -6Q2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(90 + -10Q + -3Q2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(90 + -10Q + -3Q2)' equal to zero and attempt to solve: Simplifying 90 + -10Q + -3Q2 = 0 Solving 90 + -10Q + -3Q2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -30 + 3.333333333Q + Q2 = 0 Move the constant term to the right: Add '30' to each side of the equation. -30 + 3.333333333Q + 30 + Q2 = 0 + 30 Reorder the terms: -30 + 30 + 3.333333333Q + Q2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 3.333333333Q + Q2 = 0 + 30 3.333333333Q + Q2 = 0 + 30 Combine like terms: 0 + 30 = 30 3.333333333Q + Q2 = 30 The Q term is 3.333333333Q. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333Q + 2.777777779 + Q2 = 30 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333Q + Q2 = 30 + 2.777777779 Combine like terms: 30 + 2.777777779 = 32.777777779 2.777777779 + 3.333333333Q + Q2 = 32.777777779 Factor a perfect square on the left side: (Q + 1.666666667)(Q + 1.666666667) = 32.777777779 Calculate the square root of the right side: 5.725188013 Break this problem into two subproblems by setting (Q + 1.666666667) equal to 5.725188013 and -5.725188013.Subproblem 1
Q + 1.666666667 = 5.725188013 Simplifying Q + 1.666666667 = 5.725188013 Reorder the terms: 1.666666667 + Q = 5.725188013 Solving 1.666666667 + Q = 5.725188013 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + Q = 5.725188013 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + Q = 5.725188013 + -1.666666667 Q = 5.725188013 + -1.666666667 Combine like terms: 5.725188013 + -1.666666667 = 4.058521346 Q = 4.058521346 Simplifying Q = 4.058521346Subproblem 2
Q + 1.666666667 = -5.725188013 Simplifying Q + 1.666666667 = -5.725188013 Reorder the terms: 1.666666667 + Q = -5.725188013 Solving 1.666666667 + Q = -5.725188013 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + Q = -5.725188013 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + Q = -5.725188013 + -1.666666667 Q = -5.725188013 + -1.666666667 Combine like terms: -5.725188013 + -1.666666667 = -7.39185468 Q = -7.39185468 Simplifying Q = -7.39185468Solution
The solution to the problem is based on the solutions from the subproblems. Q = {4.058521346, -7.39185468}Solution
Q = {4.058521346, -7.39185468}
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