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Simplifying 5q2 + 3q = 3q2 + -4q + 18 Reorder the terms: 3q + 5q2 = 3q2 + -4q + 18 Reorder the terms: 3q + 5q2 = 18 + -4q + 3q2 Solving 3q + 5q2 = 18 + -4q + 3q2 Solving for variable 'q'. Reorder the terms: -18 + 3q + 4q + 5q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 3q + 4q = 7q -18 + 7q + 5q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 5q2 + -3q2 = 2q2 -18 + 7q + 2q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Reorder the terms: -18 + 7q + 2q2 = 18 + -18 + -4q + 4q + 3q2 + -3q2 Combine like terms: 18 + -18 = 0 -18 + 7q + 2q2 = 0 + -4q + 4q + 3q2 + -3q2 -18 + 7q + 2q2 = -4q + 4q + 3q2 + -3q2 Combine like terms: -4q + 4q = 0 -18 + 7q + 2q2 = 0 + 3q2 + -3q2 -18 + 7q + 2q2 = 3q2 + -3q2 Combine like terms: 3q2 + -3q2 = 0 -18 + 7q + 2q2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9 + 3.5q + q2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 3.5q + 9 + q2 = 0 + 9 Reorder the terms: -9 + 9 + 3.5q + q2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 3.5q + q2 = 0 + 9 3.5q + q2 = 0 + 9 Combine like terms: 0 + 9 = 9 3.5q + q2 = 9 The q term is 3.5q. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5q + 3.0625 + q2 = 9 + 3.0625 Reorder the terms: 3.0625 + 3.5q + q2 = 9 + 3.0625 Combine like terms: 9 + 3.0625 = 12.0625 3.0625 + 3.5q + q2 = 12.0625 Factor a perfect square on the left side: (q + 1.75)(q + 1.75) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (q + 1.75) equal to 3.473110997 and -3.473110997.Subproblem 1
q + 1.75 = 3.473110997 Simplifying q + 1.75 = 3.473110997 Reorder the terms: 1.75 + q = 3.473110997 Solving 1.75 + q = 3.473110997 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + q = 3.473110997 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + q = 3.473110997 + -1.75 q = 3.473110997 + -1.75 Combine like terms: 3.473110997 + -1.75 = 1.723110997 q = 1.723110997 Simplifying q = 1.723110997Subproblem 2
q + 1.75 = -3.473110997 Simplifying q + 1.75 = -3.473110997 Reorder the terms: 1.75 + q = -3.473110997 Solving 1.75 + q = -3.473110997 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + q = -3.473110997 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + q = -3.473110997 + -1.75 q = -3.473110997 + -1.75 Combine like terms: -3.473110997 + -1.75 = -5.223110997 q = -5.223110997 Simplifying q = -5.223110997Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.723110997, -5.223110997}
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