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Simplifying 8q + -3 + 5q(2 + q) = 0 8q + -3 + (2 * 5q + q * 5q) = 0 8q + -3 + (10q + 5q2) = 0 Reorder the terms: -3 + 8q + 10q + 5q2 = 0 Combine like terms: 8q + 10q = 18q -3 + 18q + 5q2 = 0 Solving -3 + 18q + 5q2 = 0 Solving for variable 'q'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.6 + 3.6q + q2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + 3.6q + 0.6 + q2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + 3.6q + q2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + 3.6q + q2 = 0 + 0.6 3.6q + q2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 3.6q + q2 = 0.6 The q term is 3.6q. Take half its coefficient (1.8). Square it (3.24) and add it to both sides. Add '3.24' to each side of the equation. 3.6q + 3.24 + q2 = 0.6 + 3.24 Reorder the terms: 3.24 + 3.6q + q2 = 0.6 + 3.24 Combine like terms: 0.6 + 3.24 = 3.84 3.24 + 3.6q + q2 = 3.84 Factor a perfect square on the left side: (q + 1.8)(q + 1.8) = 3.84 Calculate the square root of the right side: 1.959591794 Break this problem into two subproblems by setting (q + 1.8) equal to 1.959591794 and -1.959591794.Subproblem 1
q + 1.8 = 1.959591794 Simplifying q + 1.8 = 1.959591794 Reorder the terms: 1.8 + q = 1.959591794 Solving 1.8 + q = 1.959591794 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.8' to each side of the equation. 1.8 + -1.8 + q = 1.959591794 + -1.8 Combine like terms: 1.8 + -1.8 = 0.0 0.0 + q = 1.959591794 + -1.8 q = 1.959591794 + -1.8 Combine like terms: 1.959591794 + -1.8 = 0.159591794 q = 0.159591794 Simplifying q = 0.159591794Subproblem 2
q + 1.8 = -1.959591794 Simplifying q + 1.8 = -1.959591794 Reorder the terms: 1.8 + q = -1.959591794 Solving 1.8 + q = -1.959591794 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.8' to each side of the equation. 1.8 + -1.8 + q = -1.959591794 + -1.8 Combine like terms: 1.8 + -1.8 = 0.0 0.0 + q = -1.959591794 + -1.8 q = -1.959591794 + -1.8 Combine like terms: -1.959591794 + -1.8 = -3.759591794 q = -3.759591794 Simplifying q = -3.759591794Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.159591794, -3.759591794}
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