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Simplifying 4x2 + 23x + 11 = 0 Reorder the terms: 11 + 23x + 4x2 = 0 Solving 11 + 23x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.75 + 5.75x + x2 = 0 Move the constant term to the right: Add '-2.75' to each side of the equation. 2.75 + 5.75x + -2.75 + x2 = 0 + -2.75 Reorder the terms: 2.75 + -2.75 + 5.75x + x2 = 0 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + 5.75x + x2 = 0 + -2.75 5.75x + x2 = 0 + -2.75 Combine like terms: 0 + -2.75 = -2.75 5.75x + x2 = -2.75 The x term is 5.75x. Take half its coefficient (2.875). Square it (8.265625) and add it to both sides. Add '8.265625' to each side of the equation. 5.75x + 8.265625 + x2 = -2.75 + 8.265625 Reorder the terms: 8.265625 + 5.75x + x2 = -2.75 + 8.265625 Combine like terms: -2.75 + 8.265625 = 5.515625 8.265625 + 5.75x + x2 = 5.515625 Factor a perfect square on the left side: (x + 2.875)(x + 2.875) = 5.515625 Calculate the square root of the right side: 2.348536779 Break this problem into two subproblems by setting (x + 2.875) equal to 2.348536779 and -2.348536779.Subproblem 1
x + 2.875 = 2.348536779 Simplifying x + 2.875 = 2.348536779 Reorder the terms: 2.875 + x = 2.348536779 Solving 2.875 + x = 2.348536779 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + x = 2.348536779 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + x = 2.348536779 + -2.875 x = 2.348536779 + -2.875 Combine like terms: 2.348536779 + -2.875 = -0.526463221 x = -0.526463221 Simplifying x = -0.526463221Subproblem 2
x + 2.875 = -2.348536779 Simplifying x + 2.875 = -2.348536779 Reorder the terms: 2.875 + x = -2.348536779 Solving 2.875 + x = -2.348536779 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + x = -2.348536779 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + x = -2.348536779 + -2.875 x = -2.348536779 + -2.875 Combine like terms: -2.348536779 + -2.875 = -5.223536779 x = -5.223536779 Simplifying x = -5.223536779Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.526463221, -5.223536779}
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