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Simplifying 3q2 + -16q + 12 = 0 Reorder the terms: 12 + -16q + 3q2 = 0 Solving 12 + -16q + 3q2 = 0 Solving for variable 'q'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 4 + -5.333333333q + q2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -5.333333333q + -4 + q2 = 0 + -4 Reorder the terms: 4 + -4 + -5.333333333q + q2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -5.333333333q + q2 = 0 + -4 -5.333333333q + q2 = 0 + -4 Combine like terms: 0 + -4 = -4 -5.333333333q + q2 = -4 The q term is -5.333333333q. Take half its coefficient (-2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. -5.333333333q + 7.111111113 + q2 = -4 + 7.111111113 Reorder the terms: 7.111111113 + -5.333333333q + q2 = -4 + 7.111111113 Combine like terms: -4 + 7.111111113 = 3.111111113 7.111111113 + -5.333333333q + q2 = 3.111111113 Factor a perfect square on the left side: (q + -2.666666667)(q + -2.666666667) = 3.111111113 Calculate the square root of the right side: 1.763834208 Break this problem into two subproblems by setting (q + -2.666666667) equal to 1.763834208 and -1.763834208.Subproblem 1
q + -2.666666667 = 1.763834208 Simplifying q + -2.666666667 = 1.763834208 Reorder the terms: -2.666666667 + q = 1.763834208 Solving -2.666666667 + q = 1.763834208 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + q = 1.763834208 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + q = 1.763834208 + 2.666666667 q = 1.763834208 + 2.666666667 Combine like terms: 1.763834208 + 2.666666667 = 4.430500875 q = 4.430500875 Simplifying q = 4.430500875Subproblem 2
q + -2.666666667 = -1.763834208 Simplifying q + -2.666666667 = -1.763834208 Reorder the terms: -2.666666667 + q = -1.763834208 Solving -2.666666667 + q = -1.763834208 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + q = -1.763834208 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + q = -1.763834208 + 2.666666667 q = -1.763834208 + 2.666666667 Combine like terms: -1.763834208 + 2.666666667 = 0.902832459 q = 0.902832459 Simplifying q = 0.902832459Solution
The solution to the problem is based on the solutions from the subproblems. q = {4.430500875, 0.902832459}
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