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Simplifying 6q2 + 3q = -2q + 24 Reorder the terms: 3q + 6q2 = -2q + 24 Reorder the terms: 3q + 6q2 = 24 + -2q Solving 3q + 6q2 = 24 + -2q Solving for variable 'q'. Reorder the terms: -24 + 3q + 2q + 6q2 = 24 + -2q + -24 + 2q Combine like terms: 3q + 2q = 5q -24 + 5q + 6q2 = 24 + -2q + -24 + 2q Reorder the terms: -24 + 5q + 6q2 = 24 + -24 + -2q + 2q Combine like terms: 24 + -24 = 0 -24 + 5q + 6q2 = 0 + -2q + 2q -24 + 5q + 6q2 = -2q + 2q Combine like terms: -2q + 2q = 0 -24 + 5q + 6q2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4 + 0.8333333333q + q2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.8333333333q + 4 + q2 = 0 + 4 Reorder the terms: -4 + 4 + 0.8333333333q + q2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.8333333333q + q2 = 0 + 4 0.8333333333q + q2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.8333333333q + q2 = 4 The q term is 0.8333333333q. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333q + 0.1736111111 + q2 = 4 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333q + q2 = 4 + 0.1736111111 Combine like terms: 4 + 0.1736111111 = 4.1736111111 0.1736111111 + 0.8333333333q + q2 = 4.1736111111 Factor a perfect square on the left side: (q + 0.4166666667)(q + 0.4166666667) = 4.1736111111 Calculate the square root of the right side: 2.042941779 Break this problem into two subproblems by setting (q + 0.4166666667) equal to 2.042941779 and -2.042941779.Subproblem 1
q + 0.4166666667 = 2.042941779 Simplifying q + 0.4166666667 = 2.042941779 Reorder the terms: 0.4166666667 + q = 2.042941779 Solving 0.4166666667 + q = 2.042941779 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + q = 2.042941779 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + q = 2.042941779 + -0.4166666667 q = 2.042941779 + -0.4166666667 Combine like terms: 2.042941779 + -0.4166666667 = 1.6262751123 q = 1.6262751123 Simplifying q = 1.6262751123Subproblem 2
q + 0.4166666667 = -2.042941779 Simplifying q + 0.4166666667 = -2.042941779 Reorder the terms: 0.4166666667 + q = -2.042941779 Solving 0.4166666667 + q = -2.042941779 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + q = -2.042941779 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + q = -2.042941779 + -0.4166666667 q = -2.042941779 + -0.4166666667 Combine like terms: -2.042941779 + -0.4166666667 = -2.4596084457 q = -2.4596084457 Simplifying q = -2.4596084457Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.6262751123, -2.4596084457}
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