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Simplifying 2q2 + 5q = -3q + 3 Reorder the terms: 5q + 2q2 = -3q + 3 Reorder the terms: 5q + 2q2 = 3 + -3q Solving 5q + 2q2 = 3 + -3q Solving for variable 'q'. Reorder the terms: -3 + 5q + 3q + 2q2 = 3 + -3q + -3 + 3q Combine like terms: 5q + 3q = 8q -3 + 8q + 2q2 = 3 + -3q + -3 + 3q Reorder the terms: -3 + 8q + 2q2 = 3 + -3 + -3q + 3q Combine like terms: 3 + -3 = 0 -3 + 8q + 2q2 = 0 + -3q + 3q -3 + 8q + 2q2 = -3q + 3q Combine like terms: -3q + 3q = 0 -3 + 8q + 2q2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 4q + q2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 4q + 1.5 + q2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 4q + q2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 4q + q2 = 0 + 1.5 4q + q2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 4q + q2 = 1.5 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 = 1.5 + 4 Reorder the terms: 4 + 4q + q2 = 1.5 + 4 Combine like terms: 1.5 + 4 = 5.5 4 + 4q + q2 = 5.5 Factor a perfect square on the left side: (q + 2)(q + 2) = 5.5 Calculate the square root of the right side: 2.34520788 Break this problem into two subproblems by setting (q + 2) equal to 2.34520788 and -2.34520788.Subproblem 1
q + 2 = 2.34520788 Simplifying q + 2 = 2.34520788 Reorder the terms: 2 + q = 2.34520788 Solving 2 + q = 2.34520788 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 = 2.34520788 + -2 Combine like terms: 2 + -2 = 0 0 + q = 2.34520788 + -2 q = 2.34520788 + -2 Combine like terms: 2.34520788 + -2 = 0.34520788 q = 0.34520788 Simplifying q = 0.34520788Subproblem 2
q + 2 = -2.34520788 Simplifying q + 2 = -2.34520788 Reorder the terms: 2 + q = -2.34520788 Solving 2 + q = -2.34520788 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 = -2.34520788 + -2 Combine like terms: 2 + -2 = 0 0 + q = -2.34520788 + -2 q = -2.34520788 + -2 Combine like terms: -2.34520788 + -2 = -4.34520788 q = -4.34520788 Simplifying q = -4.34520788Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.34520788, -4.34520788}
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