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Simplifying 86 + -6q + -3q2 = 0.25q2 + 10 Reorder the terms: 86 + -6q + -3q2 = 10 + 0.25q2 Solving 86 + -6q + -3q2 = 10 + 0.25q2 Solving for variable 'q'. Reorder the terms: 86 + -10 + -6q + -3q2 + -0.25q2 = 10 + 0.25q2 + -10 + -0.25q2 Combine like terms: 86 + -10 = 76 76 + -6q + -3q2 + -0.25q2 = 10 + 0.25q2 + -10 + -0.25q2 Combine like terms: -3q2 + -0.25q2 = -3.25q2 76 + -6q + -3.25q2 = 10 + 0.25q2 + -10 + -0.25q2 Reorder the terms: 76 + -6q + -3.25q2 = 10 + -10 + 0.25q2 + -0.25q2 Combine like terms: 10 + -10 = 0 76 + -6q + -3.25q2 = 0 + 0.25q2 + -0.25q2 76 + -6q + -3.25q2 = 0.25q2 + -0.25q2 Combine like terms: 0.25q2 + -0.25q2 = 0.00 76 + -6q + -3.25q2 = 0.00 Begin completing the square. Divide all terms by -3.25 the coefficient of the squared term: Divide each side by '-3.25'. -23.38461538 + 1.846153846q + q2 = 0 Move the constant term to the right: Add '23.38461538' to each side of the equation. -23.38461538 + 1.846153846q + 23.38461538 + q2 = 0 + 23.38461538 Reorder the terms: -23.38461538 + 23.38461538 + 1.846153846q + q2 = 0 + 23.38461538 Combine like terms: -23.38461538 + 23.38461538 = 0.00000000 0.00000000 + 1.846153846q + q2 = 0 + 23.38461538 1.846153846q + q2 = 0 + 23.38461538 Combine like terms: 0 + 23.38461538 = 23.38461538 1.846153846q + q2 = 23.38461538 The q term is 1.846153846q. Take half its coefficient (0.923076923). Square it (0.8520710058) and add it to both sides. Add '0.8520710058' to each side of the equation. 1.846153846q + 0.8520710058 + q2 = 23.38461538 + 0.8520710058 Reorder the terms: 0.8520710058 + 1.846153846q + q2 = 23.38461538 + 0.8520710058 Combine like terms: 23.38461538 + 0.8520710058 = 24.2366863858 0.8520710058 + 1.846153846q + q2 = 24.2366863858 Factor a perfect square on the left side: (q + 0.923076923)(q + 0.923076923) = 24.2366863858 Calculate the square root of the right side: 4.923076923 Break this problem into two subproblems by setting (q + 0.923076923) equal to 4.923076923 and -4.923076923.Subproblem 1
q + 0.923076923 = 4.923076923 Simplifying q + 0.923076923 = 4.923076923 Reorder the terms: 0.923076923 + q = 4.923076923 Solving 0.923076923 + q = 4.923076923 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.923076923' to each side of the equation. 0.923076923 + -0.923076923 + q = 4.923076923 + -0.923076923 Combine like terms: 0.923076923 + -0.923076923 = 0.000000000 0.000000000 + q = 4.923076923 + -0.923076923 q = 4.923076923 + -0.923076923 Combine like terms: 4.923076923 + -0.923076923 = 4 q = 4 Simplifying q = 4Subproblem 2
q + 0.923076923 = -4.923076923 Simplifying q + 0.923076923 = -4.923076923 Reorder the terms: 0.923076923 + q = -4.923076923 Solving 0.923076923 + q = -4.923076923 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.923076923' to each side of the equation. 0.923076923 + -0.923076923 + q = -4.923076923 + -0.923076923 Combine like terms: 0.923076923 + -0.923076923 = 0.000000000 0.000000000 + q = -4.923076923 + -0.923076923 q = -4.923076923 + -0.923076923 Combine like terms: -4.923076923 + -0.923076923 = -5.846153846 q = -5.846153846 Simplifying q = -5.846153846Solution
The solution to the problem is based on the solutions from the subproblems. q = {4, -5.846153846}
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