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Simplifying 5q2 + 32q + 12 = 23 Reorder the terms: 12 + 32q + 5q2 = 23 Solving 12 + 32q + 5q2 = 23 Solving for variable 'q'. Reorder the terms: 12 + -23 + 32q + 5q2 = 23 + -23 Combine like terms: 12 + -23 = -11 -11 + 32q + 5q2 = 23 + -23 Combine like terms: 23 + -23 = 0 -11 + 32q + 5q2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.2 + 6.4q + q2 = 0 Move the constant term to the right: Add '2.2' to each side of the equation. -2.2 + 6.4q + 2.2 + q2 = 0 + 2.2 Reorder the terms: -2.2 + 2.2 + 6.4q + q2 = 0 + 2.2 Combine like terms: -2.2 + 2.2 = 0.0 0.0 + 6.4q + q2 = 0 + 2.2 6.4q + q2 = 0 + 2.2 Combine like terms: 0 + 2.2 = 2.2 6.4q + q2 = 2.2 The q term is 6.4q. Take half its coefficient (3.2). Square it (10.24) and add it to both sides. Add '10.24' to each side of the equation. 6.4q + 10.24 + q2 = 2.2 + 10.24 Reorder the terms: 10.24 + 6.4q + q2 = 2.2 + 10.24 Combine like terms: 2.2 + 10.24 = 12.44 10.24 + 6.4q + q2 = 12.44 Factor a perfect square on the left side: (q + 3.2)(q + 3.2) = 12.44 Calculate the square root of the right side: 3.527038418 Break this problem into two subproblems by setting (q + 3.2) equal to 3.527038418 and -3.527038418.Subproblem 1
q + 3.2 = 3.527038418 Simplifying q + 3.2 = 3.527038418 Reorder the terms: 3.2 + q = 3.527038418 Solving 3.2 + q = 3.527038418 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3.2' to each side of the equation. 3.2 + -3.2 + q = 3.527038418 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + q = 3.527038418 + -3.2 q = 3.527038418 + -3.2 Combine like terms: 3.527038418 + -3.2 = 0.327038418 q = 0.327038418 Simplifying q = 0.327038418Subproblem 2
q + 3.2 = -3.527038418 Simplifying q + 3.2 = -3.527038418 Reorder the terms: 3.2 + q = -3.527038418 Solving 3.2 + q = -3.527038418 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3.2' to each side of the equation. 3.2 + -3.2 + q = -3.527038418 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + q = -3.527038418 + -3.2 q = -3.527038418 + -3.2 Combine like terms: -3.527038418 + -3.2 = -6.727038418 q = -6.727038418 Simplifying q = -6.727038418Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.327038418, -6.727038418}
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