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Simplifying -1q2 + 39.5q + -109.5 = 0 Reorder the terms: -109.5 + 39.5q + -1q2 = 0 Solving -109.5 + 39.5q + -1q2 = 0 Solving for variable 'q'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 109.5 + -39.5q + q2 = 0 Move the constant term to the right: Add '-109.5' to each side of the equation. 109.5 + -39.5q + -109.5 + q2 = 0 + -109.5 Reorder the terms: 109.5 + -109.5 + -39.5q + q2 = 0 + -109.5 Combine like terms: 109.5 + -109.5 = 0.0 0.0 + -39.5q + q2 = 0 + -109.5 -39.5q + q2 = 0 + -109.5 Combine like terms: 0 + -109.5 = -109.5 -39.5q + q2 = -109.5 The q term is -39.5q. Take half its coefficient (-19.75). Square it (390.0625) and add it to both sides. Add '390.0625' to each side of the equation. -39.5q + 390.0625 + q2 = -109.5 + 390.0625 Reorder the terms: 390.0625 + -39.5q + q2 = -109.5 + 390.0625 Combine like terms: -109.5 + 390.0625 = 280.5625 390.0625 + -39.5q + q2 = 280.5625 Factor a perfect square on the left side: (q + -19.75)(q + -19.75) = 280.5625 Calculate the square root of the right side: 16.75 Break this problem into two subproblems by setting (q + -19.75) equal to 16.75 and -16.75.Subproblem 1
q + -19.75 = 16.75 Simplifying q + -19.75 = 16.75 Reorder the terms: -19.75 + q = 16.75 Solving -19.75 + q = 16.75 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '19.75' to each side of the equation. -19.75 + 19.75 + q = 16.75 + 19.75 Combine like terms: -19.75 + 19.75 = 0.00 0.00 + q = 16.75 + 19.75 q = 16.75 + 19.75 Combine like terms: 16.75 + 19.75 = 36.5 q = 36.5 Simplifying q = 36.5Subproblem 2
q + -19.75 = -16.75 Simplifying q + -19.75 = -16.75 Reorder the terms: -19.75 + q = -16.75 Solving -19.75 + q = -16.75 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '19.75' to each side of the equation. -19.75 + 19.75 + q = -16.75 + 19.75 Combine like terms: -19.75 + 19.75 = 0.00 0.00 + q = -16.75 + 19.75 q = -16.75 + 19.75 Combine like terms: -16.75 + 19.75 = 3 q = 3 Simplifying q = 3Solution
The solution to the problem is based on the solutions from the subproblems. q = {36.5, 3}
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