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Simplifying 85 + -3q = 4q2 + 6 Reorder the terms: 85 + -3q = 6 + 4q2 Solving 85 + -3q = 6 + 4q2 Solving for variable 'q'. Reorder the terms: 85 + -6 + -3q + -4q2 = 6 + 4q2 + -6 + -4q2 Combine like terms: 85 + -6 = 79 79 + -3q + -4q2 = 6 + 4q2 + -6 + -4q2 Reorder the terms: 79 + -3q + -4q2 = 6 + -6 + 4q2 + -4q2 Combine like terms: 6 + -6 = 0 79 + -3q + -4q2 = 0 + 4q2 + -4q2 79 + -3q + -4q2 = 4q2 + -4q2 Combine like terms: 4q2 + -4q2 = 0 79 + -3q + -4q2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -19.75 + 0.75q + q2 = 0 Move the constant term to the right: Add '19.75' to each side of the equation. -19.75 + 0.75q + 19.75 + q2 = 0 + 19.75 Reorder the terms: -19.75 + 19.75 + 0.75q + q2 = 0 + 19.75 Combine like terms: -19.75 + 19.75 = 0.00 0.00 + 0.75q + q2 = 0 + 19.75 0.75q + q2 = 0 + 19.75 Combine like terms: 0 + 19.75 = 19.75 0.75q + q2 = 19.75 The q term is 0.75q. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75q + 0.140625 + q2 = 19.75 + 0.140625 Reorder the terms: 0.140625 + 0.75q + q2 = 19.75 + 0.140625 Combine like terms: 19.75 + 0.140625 = 19.890625 0.140625 + 0.75q + q2 = 19.890625 Factor a perfect square on the left side: (q + 0.375)(q + 0.375) = 19.890625 Calculate the square root of the right side: 4.459890694 Break this problem into two subproblems by setting (q + 0.375) equal to 4.459890694 and -4.459890694.Subproblem 1
q + 0.375 = 4.459890694 Simplifying q + 0.375 = 4.459890694 Reorder the terms: 0.375 + q = 4.459890694 Solving 0.375 + q = 4.459890694 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + q = 4.459890694 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + q = 4.459890694 + -0.375 q = 4.459890694 + -0.375 Combine like terms: 4.459890694 + -0.375 = 4.084890694 q = 4.084890694 Simplifying q = 4.084890694Subproblem 2
q + 0.375 = -4.459890694 Simplifying q + 0.375 = -4.459890694 Reorder the terms: 0.375 + q = -4.459890694 Solving 0.375 + q = -4.459890694 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + q = -4.459890694 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + q = -4.459890694 + -0.375 q = -4.459890694 + -0.375 Combine like terms: -4.459890694 + -0.375 = -4.834890694 q = -4.834890694 Simplifying q = -4.834890694Solution
The solution to the problem is based on the solutions from the subproblems. q = {4.084890694, -4.834890694}
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