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Simplifying q(q + 6) = 4q + 6 Reorder the terms: q(6 + q) = 4q + 6 (6 * q + q * q) = 4q + 6 (6q + q2) = 4q + 6 Reorder the terms: 6q + q2 = 6 + 4q Solving 6q + q2 = 6 + 4q Solving for variable 'q'. Reorder the terms: -6 + 6q + -4q + q2 = 6 + 4q + -6 + -4q Combine like terms: 6q + -4q = 2q -6 + 2q + q2 = 6 + 4q + -6 + -4q Reorder the terms: -6 + 2q + q2 = 6 + -6 + 4q + -4q Combine like terms: 6 + -6 = 0 -6 + 2q + q2 = 0 + 4q + -4q -6 + 2q + q2 = 4q + -4q Combine like terms: 4q + -4q = 0 -6 + 2q + q2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + 2q + 6 + q2 = 0 + 6 Reorder the terms: -6 + 6 + 2q + q2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 2q + q2 = 0 + 6 2q + q2 = 0 + 6 Combine like terms: 0 + 6 = 6 2q + q2 = 6 The q term is 2q. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2q + 1 + q2 = 6 + 1 Reorder the terms: 1 + 2q + q2 = 6 + 1 Combine like terms: 6 + 1 = 7 1 + 2q + q2 = 7 Factor a perfect square on the left side: (q + 1)(q + 1) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (q + 1) equal to 2.645751311 and -2.645751311.Subproblem 1
q + 1 = 2.645751311 Simplifying q + 1 = 2.645751311 Reorder the terms: 1 + q = 2.645751311 Solving 1 + q = 2.645751311 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + q = 2.645751311 + -1 Combine like terms: 1 + -1 = 0 0 + q = 2.645751311 + -1 q = 2.645751311 + -1 Combine like terms: 2.645751311 + -1 = 1.645751311 q = 1.645751311 Simplifying q = 1.645751311Subproblem 2
q + 1 = -2.645751311 Simplifying q + 1 = -2.645751311 Reorder the terms: 1 + q = -2.645751311 Solving 1 + q = -2.645751311 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + q = -2.645751311 + -1 Combine like terms: 1 + -1 = 0 0 + q = -2.645751311 + -1 q = -2.645751311 + -1 Combine like terms: -2.645751311 + -1 = -3.645751311 q = -3.645751311 Simplifying q = -3.645751311Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.645751311, -3.645751311}
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