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Simplifying 4q2 + q = 3q2 + -4q + 18 Reorder the terms: q + 4q2 = 3q2 + -4q + 18 Reorder the terms: q + 4q2 = 18 + -4q + 3q2 Solving q + 4q2 = 18 + -4q + 3q2 Solving for variable 'q'. Reorder the terms: -18 + q + 4q + 4q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: q + 4q = 5q -18 + 5q + 4q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 4q2 + -3q2 = 1q2 -18 + 5q + 1q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Reorder the terms: -18 + 5q + 1q2 = 18 + -18 + -4q + 4q + 3q2 + -3q2 Combine like terms: 18 + -18 = 0 -18 + 5q + 1q2 = 0 + -4q + 4q + 3q2 + -3q2 -18 + 5q + 1q2 = -4q + 4q + 3q2 + -3q2 Combine like terms: -4q + 4q = 0 -18 + 5q + 1q2 = 0 + 3q2 + -3q2 -18 + 5q + 1q2 = 3q2 + -3q2 Combine like terms: 3q2 + -3q2 = 0 -18 + 5q + 1q2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 5q + 18 + q2 = 0 + 18 Reorder the terms: -18 + 18 + 5q + q2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 5q + q2 = 0 + 18 5q + q2 = 0 + 18 Combine like terms: 0 + 18 = 18 5q + q2 = 18 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 = 18 + 6.25 Reorder the terms: 6.25 + 5q + q2 = 18 + 6.25 Combine like terms: 18 + 6.25 = 24.25 6.25 + 5q + q2 = 24.25 Factor a perfect square on the left side: (q + 2.5)(q + 2.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (q + 2.5) equal to 4.924428901 and -4.924428901.Subproblem 1
q + 2.5 = 4.924428901 Simplifying q + 2.5 = 4.924428901 Reorder the terms: 2.5 + q = 4.924428901 Solving 2.5 + q = 4.924428901 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 = 4.924428901 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + q = 4.924428901 + -2.5 q = 4.924428901 + -2.5 Combine like terms: 4.924428901 + -2.5 = 2.424428901 q = 2.424428901 Simplifying q = 2.424428901Subproblem 2
q + 2.5 = -4.924428901 Simplifying q + 2.5 = -4.924428901 Reorder the terms: 2.5 + q = -4.924428901 Solving 2.5 + q = -4.924428901 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 = -4.924428901 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + q = -4.924428901 + -2.5 q = -4.924428901 + -2.5 Combine like terms: -4.924428901 + -2.5 = -7.424428901 q = -7.424428901 Simplifying q = -7.424428901Solution
The solution to the problem is based on the solutions from the subproblems. q = {2.424428901, -7.424428901}
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