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Simplifying q2 + 2q = 1 Reorder the terms: 2q + q2 = 1 Solving 2q + q2 = 1 Solving for variable 'q'. Reorder the terms: -1 + 2q + q2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 2q + q2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 2q + 1 + q2 = 0 + 1 Reorder the terms: -1 + 1 + 2q + q2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2q + q2 = 0 + 1 2q + q2 = 0 + 1 Combine like terms: 0 + 1 = 1 2q + q2 = 1 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 = 1 + 1 Reorder the terms: 1 + 2q + q2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + 2q + q2 = 2 Factor a perfect square on the left side: (q + 1)(q + 1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (q + 1) equal to 1.414213562 and -1.414213562.Subproblem 1
q + 1 = 1.414213562 Simplifying q + 1 = 1.414213562 Reorder the terms: 1 + q = 1.414213562 Solving 1 + q = 1.414213562 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 = 1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + q = 1.414213562 + -1 q = 1.414213562 + -1 Combine like terms: 1.414213562 + -1 = 0.414213562 q = 0.414213562 Simplifying q = 0.414213562Subproblem 2
q + 1 = -1.414213562 Simplifying q + 1 = -1.414213562 Reorder the terms: 1 + q = -1.414213562 Solving 1 + q = -1.414213562 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 = -1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + q = -1.414213562 + -1 q = -1.414213562 + -1 Combine like terms: -1.414213562 + -1 = -2.414213562 q = -2.414213562 Simplifying q = -2.414213562Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.414213562, -2.414213562}
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