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Simplifying q2 + 5q + -102 = 0 Reorder the terms: -102 + 5q + q2 = 0 Solving -102 + 5q + q2 = 0 Solving for variable 'q'. Begin completing the square. Move the constant term to the right: Add '102' to each side of the equation. -102 + 5q + 102 + q2 = 0 + 102 Reorder the terms: -102 + 102 + 5q + q2 = 0 + 102 Combine like terms: -102 + 102 = 0 0 + 5q + q2 = 0 + 102 5q + q2 = 0 + 102 Combine like terms: 0 + 102 = 102 5q + q2 = 102 The q term is 5q. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5q + 6.25 + q2 = 102 + 6.25 Reorder the terms: 6.25 + 5q + q2 = 102 + 6.25 Combine like terms: 102 + 6.25 = 108.25 6.25 + 5q + q2 = 108.25 Factor a perfect square on the left side: (q + 2.5)(q + 2.5) = 108.25 Calculate the square root of the right side: 10.404326023 Break this problem into two subproblems by setting (q + 2.5) equal to 10.404326023 and -10.404326023.Subproblem 1
q + 2.5 = 10.404326023 Simplifying q + 2.5 = 10.404326023 Reorder the terms: 2.5 + q = 10.404326023 Solving 2.5 + q = 10.404326023 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + q = 10.404326023 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + q = 10.404326023 + -2.5 q = 10.404326023 + -2.5 Combine like terms: 10.404326023 + -2.5 = 7.904326023 q = 7.904326023 Simplifying q = 7.904326023Subproblem 2
q + 2.5 = -10.404326023 Simplifying q + 2.5 = -10.404326023 Reorder the terms: 2.5 + q = -10.404326023 Solving 2.5 + q = -10.404326023 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + q = -10.404326023 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + q = -10.404326023 + -2.5 q = -10.404326023 + -2.5 Combine like terms: -10.404326023 + -2.5 = -12.904326023 q = -12.904326023 Simplifying q = -12.904326023Solution
The solution to the problem is based on the solutions from the subproblems. q = {7.904326023, -12.904326023}
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