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Simplifying 9x2 + 23x + 11 = 0 Reorder the terms: 11 + 23x + 9x2 = 0 Solving 11 + 23x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 1.222222222 + 2.555555556x + x2 = 0 Move the constant term to the right: Add '-1.222222222' to each side of the equation. 1.222222222 + 2.555555556x + -1.222222222 + x2 = 0 + -1.222222222 Reorder the terms: 1.222222222 + -1.222222222 + 2.555555556x + x2 = 0 + -1.222222222 Combine like terms: 1.222222222 + -1.222222222 = 0.000000000 0.000000000 + 2.555555556x + x2 = 0 + -1.222222222 2.555555556x + x2 = 0 + -1.222222222 Combine like terms: 0 + -1.222222222 = -1.222222222 2.555555556x + x2 = -1.222222222 The x term is 2.555555556x. Take half its coefficient (1.277777778). Square it (1.632716050) and add it to both sides. Add '1.632716050' to each side of the equation. 2.555555556x + 1.632716050 + x2 = -1.222222222 + 1.632716050 Reorder the terms: 1.632716050 + 2.555555556x + x2 = -1.222222222 + 1.632716050 Combine like terms: -1.222222222 + 1.632716050 = 0.410493828 1.632716050 + 2.555555556x + x2 = 0.410493828 Factor a perfect square on the left side: (x + 1.277777778)(x + 1.277777778) = 0.410493828 Calculate the square root of the right side: 0.640697923 Break this problem into two subproblems by setting (x + 1.277777778) equal to 0.640697923 and -0.640697923.Subproblem 1
x + 1.277777778 = 0.640697923 Simplifying x + 1.277777778 = 0.640697923 Reorder the terms: 1.277777778 + x = 0.640697923 Solving 1.277777778 + x = 0.640697923 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.277777778' to each side of the equation. 1.277777778 + -1.277777778 + x = 0.640697923 + -1.277777778 Combine like terms: 1.277777778 + -1.277777778 = 0.000000000 0.000000000 + x = 0.640697923 + -1.277777778 x = 0.640697923 + -1.277777778 Combine like terms: 0.640697923 + -1.277777778 = -0.637079855 x = -0.637079855 Simplifying x = -0.637079855Subproblem 2
x + 1.277777778 = -0.640697923 Simplifying x + 1.277777778 = -0.640697923 Reorder the terms: 1.277777778 + x = -0.640697923 Solving 1.277777778 + x = -0.640697923 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.277777778' to each side of the equation. 1.277777778 + -1.277777778 + x = -0.640697923 + -1.277777778 Combine like terms: 1.277777778 + -1.277777778 = 0.000000000 0.000000000 + x = -0.640697923 + -1.277777778 x = -0.640697923 + -1.277777778 Combine like terms: -0.640697923 + -1.277777778 = -1.918475701 x = -1.918475701 Simplifying x = -1.918475701Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.637079855, -1.918475701}
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