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Simplifying x2 + 11x + 23 = 0 Reorder the terms: 23 + 11x + x2 = 0 Solving 23 + 11x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-23' to each side of the equation. 23 + 11x + -23 + x2 = 0 + -23 Reorder the terms: 23 + -23 + 11x + x2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + 11x + x2 = 0 + -23 11x + x2 = 0 + -23 Combine like terms: 0 + -23 = -23 11x + x2 = -23 The x term is 11x. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11x + 30.25 + x2 = -23 + 30.25 Reorder the terms: 30.25 + 11x + x2 = -23 + 30.25 Combine like terms: -23 + 30.25 = 7.25 30.25 + 11x + x2 = 7.25 Factor a perfect square on the left side: (x + 5.5)(x + 5.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (x + 5.5) equal to 2.692582404 and -2.692582404.Subproblem 1
x + 5.5 = 2.692582404 Simplifying x + 5.5 = 2.692582404 Reorder the terms: 5.5 + x = 2.692582404 Solving 5.5 + x = 2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = 2.692582404 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = 2.692582404 + -5.5 x = 2.692582404 + -5.5 Combine like terms: 2.692582404 + -5.5 = -2.807417596 x = -2.807417596 Simplifying x = -2.807417596Subproblem 2
x + 5.5 = -2.692582404 Simplifying x + 5.5 = -2.692582404 Reorder the terms: 5.5 + x = -2.692582404 Solving 5.5 + x = -2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = -2.692582404 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = -2.692582404 + -5.5 x = -2.692582404 + -5.5 Combine like terms: -2.692582404 + -5.5 = -8.192582404 x = -8.192582404 Simplifying x = -8.192582404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.807417596, -8.192582404}
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