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Simplifying 4q2 + 3 = 3q2 + -4q + 18 Reorder the terms: 3 + 4q2 = 3q2 + -4q + 18 Reorder the terms: 3 + 4q2 = 18 + -4q + 3q2 Solving 3 + 4q2 = 18 + -4q + 3q2 Solving for variable 'q'. Reorder the terms: 3 + -18 + 4q + 4q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 3 + -18 = -15 -15 + 4q + 4q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 4q2 + -3q2 = 1q2 -15 + 4q + 1q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Reorder the terms: -15 + 4q + 1q2 = 18 + -18 + -4q + 4q + 3q2 + -3q2 Combine like terms: 18 + -18 = 0 -15 + 4q + 1q2 = 0 + -4q + 4q + 3q2 + -3q2 -15 + 4q + 1q2 = -4q + 4q + 3q2 + -3q2 Combine like terms: -4q + 4q = 0 -15 + 4q + 1q2 = 0 + 3q2 + -3q2 -15 + 4q + 1q2 = 3q2 + -3q2 Combine like terms: 3q2 + -3q2 = 0 -15 + 4q + 1q2 = 0 Begin completing the square. Move the constant term to the right: Add '15' to each side of the equation. -15 + 4q + 15 + q2 = 0 + 15 Reorder the terms: -15 + 15 + 4q + q2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 4q + q2 = 0 + 15 4q + q2 = 0 + 15 Combine like terms: 0 + 15 = 15 4q + q2 = 15 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 = 15 + 4 Reorder the terms: 4 + 4q + q2 = 15 + 4 Combine like terms: 15 + 4 = 19 4 + 4q + q2 = 19 Factor a perfect square on the left side: (q + 2)(q + 2) = 19 Calculate the square root of the right side: 4.358898944 Break this problem into two subproblems by setting (q + 2) equal to 4.358898944 and -4.358898944.Subproblem 1
q + 2 = 4.358898944 Simplifying q + 2 = 4.358898944 Reorder the terms: 2 + q = 4.358898944 Solving 2 + q = 4.358898944 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 = 4.358898944 + -2 Combine like terms: 2 + -2 = 0 0 + q = 4.358898944 + -2 q = 4.358898944 + -2 Combine like terms: 4.358898944 + -2 = 2.358898944 q = 2.358898944 Simplifying q = 2.358898944Subproblem 2
q + 2 = -4.358898944 Simplifying q + 2 = -4.358898944 Reorder the terms: 2 + q = -4.358898944 Solving 2 + q = -4.358898944 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 = -4.358898944 + -2 Combine like terms: 2 + -2 = 0 0 + q = -4.358898944 + -2 q = -4.358898944 + -2 Combine like terms: -4.358898944 + -2 = -6.358898944 q = -6.358898944 Simplifying q = -6.358898944Solution
The solution to the problem is based on the solutions from the subproblems. q = {2.358898944, -6.358898944}
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