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Simplifying 400 + -50 = 3q2 + 6q Combine like terms: 400 + -50 = 350 350 = 3q2 + 6q Reorder the terms: 350 = 6q + 3q2 Solving 350 = 6q + 3q2 Solving for variable 'q'. Reorder the terms: 350 + -6q + -3q2 = 6q + -6q + 3q2 + -3q2 Combine like terms: 6q + -6q = 0 350 + -6q + -3q2 = 0 + 3q2 + -3q2 350 + -6q + -3q2 = 3q2 + -3q2 Combine like terms: 3q2 + -3q2 = 0 350 + -6q + -3q2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -116.6666667 + 2q + q2 = 0 Move the constant term to the right: Add '116.6666667' to each side of the equation. -116.6666667 + 2q + 116.6666667 + q2 = 0 + 116.6666667 Reorder the terms: -116.6666667 + 116.6666667 + 2q + q2 = 0 + 116.6666667 Combine like terms: -116.6666667 + 116.6666667 = 0.0000000 0.0000000 + 2q + q2 = 0 + 116.6666667 2q + q2 = 0 + 116.6666667 Combine like terms: 0 + 116.6666667 = 116.6666667 2q + q2 = 116.6666667 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 = 116.6666667 + 1 Reorder the terms: 1 + 2q + q2 = 116.6666667 + 1 Combine like terms: 116.6666667 + 1 = 117.6666667 1 + 2q + q2 = 117.6666667 Factor a perfect square on the left side: (q + 1)(q + 1) = 117.6666667 Calculate the square root of the right side: 10.847426732 Break this problem into two subproblems by setting (q + 1) equal to 10.847426732 and -10.847426732.Subproblem 1
q + 1 = 10.847426732 Simplifying q + 1 = 10.847426732 Reorder the terms: 1 + q = 10.847426732 Solving 1 + q = 10.847426732 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 = 10.847426732 + -1 Combine like terms: 1 + -1 = 0 0 + q = 10.847426732 + -1 q = 10.847426732 + -1 Combine like terms: 10.847426732 + -1 = 9.847426732 q = 9.847426732 Simplifying q = 9.847426732Subproblem 2
q + 1 = -10.847426732 Simplifying q + 1 = -10.847426732 Reorder the terms: 1 + q = -10.847426732 Solving 1 + q = -10.847426732 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 = -10.847426732 + -1 Combine like terms: 1 + -1 = 0 0 + q = -10.847426732 + -1 q = -10.847426732 + -1 Combine like terms: -10.847426732 + -1 = -11.847426732 q = -11.847426732 Simplifying q = -11.847426732Solution
The solution to the problem is based on the solutions from the subproblems. q = {9.847426732, -11.847426732}
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