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Simplifying 35 = 10 + 2Q + 0.5Q2 Solving 35 = 10 + 2Q + 0.5Q2 Solving for variable 'Q'. Combine like terms: 35 + -10 = 25 25 + -2Q + -0.5Q2 = 10 + 2Q + 0.5Q2 + -10 + -2Q + -0.5Q2 Reorder the terms: 25 + -2Q + -0.5Q2 = 10 + -10 + 2Q + -2Q + 0.5Q2 + -0.5Q2 Combine like terms: 10 + -10 = 0 25 + -2Q + -0.5Q2 = 0 + 2Q + -2Q + 0.5Q2 + -0.5Q2 25 + -2Q + -0.5Q2 = 2Q + -2Q + 0.5Q2 + -0.5Q2 Combine like terms: 2Q + -2Q = 0 25 + -2Q + -0.5Q2 = 0 + 0.5Q2 + -0.5Q2 25 + -2Q + -0.5Q2 = 0.5Q2 + -0.5Q2 Combine like terms: 0.5Q2 + -0.5Q2 = 0.0 25 + -2Q + -0.5Q2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -50 + 4Q + Q2 = 0 Move the constant term to the right: Add '50' to each side of the equation. -50 + 4Q + 50 + Q2 = 0 + 50 Reorder the terms: -50 + 50 + 4Q + Q2 = 0 + 50 Combine like terms: -50 + 50 = 0 0 + 4Q + Q2 = 0 + 50 4Q + Q2 = 0 + 50 Combine like terms: 0 + 50 = 50 4Q + Q2 = 50 The Q term is 4Q. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4Q + 4 + Q2 = 50 + 4 Reorder the terms: 4 + 4Q + Q2 = 50 + 4 Combine like terms: 50 + 4 = 54 4 + 4Q + Q2 = 54 Factor a perfect square on the left side: (Q + 2)(Q + 2) = 54 Calculate the square root of the right side: 7.348469228 Break this problem into two subproblems by setting (Q + 2) equal to 7.348469228 and -7.348469228.Subproblem 1
Q + 2 = 7.348469228 Simplifying Q + 2 = 7.348469228 Reorder the terms: 2 + Q = 7.348469228 Solving 2 + Q = 7.348469228 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + Q = 7.348469228 + -2 Combine like terms: 2 + -2 = 0 0 + Q = 7.348469228 + -2 Q = 7.348469228 + -2 Combine like terms: 7.348469228 + -2 = 5.348469228 Q = 5.348469228 Simplifying Q = 5.348469228Subproblem 2
Q + 2 = -7.348469228 Simplifying Q + 2 = -7.348469228 Reorder the terms: 2 + Q = -7.348469228 Solving 2 + Q = -7.348469228 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + Q = -7.348469228 + -2 Combine like terms: 2 + -2 = 0 0 + Q = -7.348469228 + -2 Q = -7.348469228 + -2 Combine like terms: -7.348469228 + -2 = -9.348469228 Q = -9.348469228 Simplifying Q = -9.348469228Solution
The solution to the problem is based on the solutions from the subproblems. Q = {5.348469228, -9.348469228}
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