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Simplifying x2 + 23x + 19 = 0 Reorder the terms: 19 + 23x + x2 = 0 Solving 19 + 23x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-19' to each side of the equation. 19 + 23x + -19 + x2 = 0 + -19 Reorder the terms: 19 + -19 + 23x + x2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + 23x + x2 = 0 + -19 23x + x2 = 0 + -19 Combine like terms: 0 + -19 = -19 23x + x2 = -19 The x term is 23x. Take half its coefficient (11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. 23x + 132.25 + x2 = -19 + 132.25 Reorder the terms: 132.25 + 23x + x2 = -19 + 132.25 Combine like terms: -19 + 132.25 = 113.25 132.25 + 23x + x2 = 113.25 Factor a perfect square on the left side: (x + 11.5)(x + 11.5) = 113.25 Calculate the square root of the right side: 10.641898327 Break this problem into two subproblems by setting (x + 11.5) equal to 10.641898327 and -10.641898327.Subproblem 1
x + 11.5 = 10.641898327 Simplifying x + 11.5 = 10.641898327 Reorder the terms: 11.5 + x = 10.641898327 Solving 11.5 + x = 10.641898327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + x = 10.641898327 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + x = 10.641898327 + -11.5 x = 10.641898327 + -11.5 Combine like terms: 10.641898327 + -11.5 = -0.858101673 x = -0.858101673 Simplifying x = -0.858101673Subproblem 2
x + 11.5 = -10.641898327 Simplifying x + 11.5 = -10.641898327 Reorder the terms: 11.5 + x = -10.641898327 Solving 11.5 + x = -10.641898327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + x = -10.641898327 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + x = -10.641898327 + -11.5 x = -10.641898327 + -11.5 Combine like terms: -10.641898327 + -11.5 = -22.141898327 x = -22.141898327 Simplifying x = -22.141898327Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.858101673, -22.141898327}
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