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Simplifying 2q2 + -9q + 5 = -3 Reorder the terms: 5 + -9q + 2q2 = -3 Solving 5 + -9q + 2q2 = -3 Solving for variable 'q'. Reorder the terms: 5 + 3 + -9q + 2q2 = -3 + 3 Combine like terms: 5 + 3 = 8 8 + -9q + 2q2 = -3 + 3 Combine like terms: -3 + 3 = 0 8 + -9q + 2q2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 4 + -4.5q + q2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -4.5q + -4 + q2 = 0 + -4 Reorder the terms: 4 + -4 + -4.5q + q2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -4.5q + q2 = 0 + -4 -4.5q + q2 = 0 + -4 Combine like terms: 0 + -4 = -4 -4.5q + q2 = -4 The q term is -4.5q. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5q + 5.0625 + q2 = -4 + 5.0625 Reorder the terms: 5.0625 + -4.5q + q2 = -4 + 5.0625 Combine like terms: -4 + 5.0625 = 1.0625 5.0625 + -4.5q + q2 = 1.0625 Factor a perfect square on the left side: (q + -2.25)(q + -2.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (q + -2.25) equal to 1.030776406 and -1.030776406.Subproblem 1
q + -2.25 = 1.030776406 Simplifying q + -2.25 = 1.030776406 Reorder the terms: -2.25 + q = 1.030776406 Solving -2.25 + q = 1.030776406 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + q = 1.030776406 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + q = 1.030776406 + 2.25 q = 1.030776406 + 2.25 Combine like terms: 1.030776406 + 2.25 = 3.280776406 q = 3.280776406 Simplifying q = 3.280776406Subproblem 2
q + -2.25 = -1.030776406 Simplifying q + -2.25 = -1.030776406 Reorder the terms: -2.25 + q = -1.030776406 Solving -2.25 + q = -1.030776406 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + q = -1.030776406 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + q = -1.030776406 + 2.25 q = -1.030776406 + 2.25 Combine like terms: -1.030776406 + 2.25 = 1.219223594 q = 1.219223594 Simplifying q = 1.219223594Solution
The solution to the problem is based on the solutions from the subproblems. q = {3.280776406, 1.219223594}
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