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Simplifying -1.5Q2 + 24Q + -22.5 = 0 Reorder the terms: -22.5 + 24Q + -1.5Q2 = 0 Solving -22.5 + 24Q + -1.5Q2 = 0 Solving for variable 'Q'. Begin completing the square. Divide all terms by -1.5 the coefficient of the squared term: Divide each side by '-1.5'. 15 + -16Q + Q2 = 0 Move the constant term to the right: Add '-15' to each side of the equation. 15 + -16Q + -15 + Q2 = 0 + -15 Reorder the terms: 15 + -15 + -16Q + Q2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + -16Q + Q2 = 0 + -15 -16Q + Q2 = 0 + -15 Combine like terms: 0 + -15 = -15 -16Q + Q2 = -15 The Q term is -16Q. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16Q + 64 + Q2 = -15 + 64 Reorder the terms: 64 + -16Q + Q2 = -15 + 64 Combine like terms: -15 + 64 = 49 64 + -16Q + Q2 = 49 Factor a perfect square on the left side: (Q + -8)(Q + -8) = 49 Calculate the square root of the right side: 7 Break this problem into two subproblems by setting (Q + -8) equal to 7 and -7.Subproblem 1
Q + -8 = 7 Simplifying Q + -8 = 7 Reorder the terms: -8 + Q = 7 Solving -8 + Q = 7 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + Q = 7 + 8 Combine like terms: -8 + 8 = 0 0 + Q = 7 + 8 Q = 7 + 8 Combine like terms: 7 + 8 = 15 Q = 15 Simplifying Q = 15Subproblem 2
Q + -8 = -7 Simplifying Q + -8 = -7 Reorder the terms: -8 + Q = -7 Solving -8 + Q = -7 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + Q = -7 + 8 Combine like terms: -8 + 8 = 0 0 + Q = -7 + 8 Q = -7 + 8 Combine like terms: -7 + 8 = 1 Q = 1 Simplifying Q = 1Solution
The solution to the problem is based on the solutions from the subproblems. Q = {15, 1}
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