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Simplifying 42.4 = 2Q2 + Q Reorder the terms: 42.4 = Q + 2Q2 Solving 42.4 = Q + 2Q2 Solving for variable 'Q'. Reorder the terms: 42.4 + -1Q + -2Q2 = Q + -1Q + 2Q2 + -2Q2 Combine like terms: Q + -1Q = 0 42.4 + -1Q + -2Q2 = 0 + 2Q2 + -2Q2 42.4 + -1Q + -2Q2 = 2Q2 + -2Q2 Combine like terms: 2Q2 + -2Q2 = 0 42.4 + -1Q + -2Q2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -21.2 + 0.5Q + Q2 = 0 Move the constant term to the right: Add '21.2' to each side of the equation. -21.2 + 0.5Q + 21.2 + Q2 = 0 + 21.2 Reorder the terms: -21.2 + 21.2 + 0.5Q + Q2 = 0 + 21.2 Combine like terms: -21.2 + 21.2 = 0.0 0.0 + 0.5Q + Q2 = 0 + 21.2 0.5Q + Q2 = 0 + 21.2 Combine like terms: 0 + 21.2 = 21.2 0.5Q + Q2 = 21.2 The Q term is 0.5Q. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5Q + 0.0625 + Q2 = 21.2 + 0.0625 Reorder the terms: 0.0625 + 0.5Q + Q2 = 21.2 + 0.0625 Combine like terms: 21.2 + 0.0625 = 21.2625 0.0625 + 0.5Q + Q2 = 21.2625 Factor a perfect square on the left side: (Q + 0.25)(Q + 0.25) = 21.2625 Calculate the square root of the right side: 4.611127845 Break this problem into two subproblems by setting (Q + 0.25) equal to 4.611127845 and -4.611127845.Subproblem 1
Q + 0.25 = 4.611127845 Simplifying Q + 0.25 = 4.611127845 Reorder the terms: 0.25 + Q = 4.611127845 Solving 0.25 + Q = 4.611127845 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + Q = 4.611127845 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + Q = 4.611127845 + -0.25 Q = 4.611127845 + -0.25 Combine like terms: 4.611127845 + -0.25 = 4.361127845 Q = 4.361127845 Simplifying Q = 4.361127845Subproblem 2
Q + 0.25 = -4.611127845 Simplifying Q + 0.25 = -4.611127845 Reorder the terms: 0.25 + Q = -4.611127845 Solving 0.25 + Q = -4.611127845 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + Q = -4.611127845 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + Q = -4.611127845 + -0.25 Q = -4.611127845 + -0.25 Combine like terms: -4.611127845 + -0.25 = -4.861127845 Q = -4.861127845 Simplifying Q = -4.861127845Solution
The solution to the problem is based on the solutions from the subproblems. Q = {4.361127845, -4.861127845}
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