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Simplifying (12q2 + 4q + 2) + -1[(-5q2 + 9q + 3) + (-20q2 + 16q + 8)] = 0 Reorder the terms: (2 + 4q + 12q2) + -1[(-5q2 + 9q + 3) + (-20q2 + 16q + 8)] = 0 Remove parenthesis around (2 + 4q + 12q2) 2 + 4q + 12q2 + -1[(-5q2 + 9q + 3) + (-20q2 + 16q + 8)] = 0 Reorder the terms: 2 + 4q + 12q2 + -1[(3 + 9q + -5q2) + (-20q2 + 16q + 8)] = 0 Remove parenthesis around (3 + 9q + -5q2) 2 + 4q + 12q2 + -1[3 + 9q + -5q2 + (-20q2 + 16q + 8)] = 0 Reorder the terms: 2 + 4q + 12q2 + -1[3 + 9q + -5q2 + (8 + 16q + -20q2)] = 0 Remove parenthesis around (8 + 16q + -20q2) 2 + 4q + 12q2 + -1[3 + 9q + -5q2 + 8 + 16q + -20q2] = 0 Reorder the terms: 2 + 4q + 12q2 + -1[3 + 8 + 9q + 16q + -5q2 + -20q2] = 0 Combine like terms: 3 + 8 = 11 2 + 4q + 12q2 + -1[11 + 9q + 16q + -5q2 + -20q2] = 0 Combine like terms: 9q + 16q = 25q 2 + 4q + 12q2 + -1[11 + 25q + -5q2 + -20q2] = 0 Combine like terms: -5q2 + -20q2 = -25q2 2 + 4q + 12q2 + -1[11 + 25q + -25q2] = 0 2 + 4q + 12q2 + [11 * -1 + 25q * -1 + -25q2 * -1] = 0 2 + 4q + 12q2 + [-11 + -25q + 25q2] = 0 Reorder the terms: 2 + -11 + 4q + -25q + 12q2 + 25q2 = 0 Combine like terms: 2 + -11 = -9 -9 + 4q + -25q + 12q2 + 25q2 = 0 Combine like terms: 4q + -25q = -21q -9 + -21q + 12q2 + 25q2 = 0 Combine like terms: 12q2 + 25q2 = 37q2 -9 + -21q + 37q2 = 0 Solving -9 + -21q + 37q2 = 0 Solving for variable 'q'. Begin completing the square. Divide all terms by 37 the coefficient of the squared term: Divide each side by '37'. -0.2432432432 + -0.5675675676q + q2 = 0 Move the constant term to the right: Add '0.2432432432' to each side of the equation. -0.2432432432 + -0.5675675676q + 0.2432432432 + q2 = 0 + 0.2432432432 Reorder the terms: -0.2432432432 + 0.2432432432 + -0.5675675676q + q2 = 0 + 0.2432432432 Combine like terms: -0.2432432432 + 0.2432432432 = 0.0000000000 0.0000000000 + -0.5675675676q + q2 = 0 + 0.2432432432 -0.5675675676q + q2 = 0 + 0.2432432432 Combine like terms: 0 + 0.2432432432 = 0.2432432432 -0.5675675676q + q2 = 0.2432432432 The q term is -0.5675675676q. Take half its coefficient (-0.2837837838). Square it (0.08053323595) and add it to both sides. Add '0.08053323595' to each side of the equation. -0.5675675676q + 0.08053323595 + q2 = 0.2432432432 + 0.08053323595 Reorder the terms: 0.08053323595 + -0.5675675676q + q2 = 0.2432432432 + 0.08053323595 Combine like terms: 0.2432432432 + 0.08053323595 = 0.32377647915 0.08053323595 + -0.5675675676q + q2 = 0.32377647915 Factor a perfect square on the left side: (q + -0.2837837838)(q + -0.2837837838) = 0.32377647915 Calculate the square root of the right side: 0.569013602 Break this problem into two subproblems by setting (q + -0.2837837838) equal to 0.569013602 and -0.569013602.Subproblem 1
q + -0.2837837838 = 0.569013602 Simplifying q + -0.2837837838 = 0.569013602 Reorder the terms: -0.2837837838 + q = 0.569013602 Solving -0.2837837838 + q = 0.569013602 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2837837838' to each side of the equation. -0.2837837838 + 0.2837837838 + q = 0.569013602 + 0.2837837838 Combine like terms: -0.2837837838 + 0.2837837838 = 0.0000000000 0.0000000000 + q = 0.569013602 + 0.2837837838 q = 0.569013602 + 0.2837837838 Combine like terms: 0.569013602 + 0.2837837838 = 0.8527973858 q = 0.8527973858 Simplifying q = 0.8527973858Subproblem 2
q + -0.2837837838 = -0.569013602 Simplifying q + -0.2837837838 = -0.569013602 Reorder the terms: -0.2837837838 + q = -0.569013602 Solving -0.2837837838 + q = -0.569013602 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2837837838' to each side of the equation. -0.2837837838 + 0.2837837838 + q = -0.569013602 + 0.2837837838 Combine like terms: -0.2837837838 + 0.2837837838 = 0.0000000000 0.0000000000 + q = -0.569013602 + 0.2837837838 q = -0.569013602 + 0.2837837838 Combine like terms: -0.569013602 + 0.2837837838 = -0.2852298182 q = -0.2852298182 Simplifying q = -0.2852298182Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.8527973858, -0.2852298182}
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