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Simplifying -3q2 + 12q = -69 Reorder the terms: 12q + -3q2 = -69 Solving 12q + -3q2 = -69 Solving for variable 'q'. Reorder the terms: 69 + 12q + -3q2 = -69 + 69 Combine like terms: -69 + 69 = 0 69 + 12q + -3q2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(23 + 4q + -1q2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(23 + 4q + -1q2)' equal to zero and attempt to solve: Simplifying 23 + 4q + -1q2 = 0 Solving 23 + 4q + -1q2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -23 + -4q + q2 = 0 Move the constant term to the right: Add '23' to each side of the equation. -23 + -4q + 23 + q2 = 0 + 23 Reorder the terms: -23 + 23 + -4q + q2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + -4q + q2 = 0 + 23 -4q + q2 = 0 + 23 Combine like terms: 0 + 23 = 23 -4q + q2 = 23 The q term is -4q. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4q + 4 + q2 = 23 + 4 Reorder the terms: 4 + -4q + q2 = 23 + 4 Combine like terms: 23 + 4 = 27 4 + -4q + q2 = 27 Factor a perfect square on the left side: (q + -2)(q + -2) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (q + -2) equal to 5.196152423 and -5.196152423.Subproblem 1
q + -2 = 5.196152423 Simplifying q + -2 = 5.196152423 Reorder the terms: -2 + q = 5.196152423 Solving -2 + q = 5.196152423 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + q = 5.196152423 + 2 Combine like terms: -2 + 2 = 0 0 + q = 5.196152423 + 2 q = 5.196152423 + 2 Combine like terms: 5.196152423 + 2 = 7.196152423 q = 7.196152423 Simplifying q = 7.196152423Subproblem 2
q + -2 = -5.196152423 Simplifying q + -2 = -5.196152423 Reorder the terms: -2 + q = -5.196152423 Solving -2 + q = -5.196152423 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + q = -5.196152423 + 2 Combine like terms: -2 + 2 = 0 0 + q = -5.196152423 + 2 q = -5.196152423 + 2 Combine like terms: -5.196152423 + 2 = -3.196152423 q = -3.196152423 Simplifying q = -3.196152423Solution
The solution to the problem is based on the solutions from the subproblems. q = {7.196152423, -3.196152423}Solution
q = {7.196152423, -3.196152423}
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