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Simplifying 11x(x + 2) + 7x + 3 = 0 Reorder the terms: 11x(2 + x) + 7x + 3 = 0 (2 * 11x + x * 11x) + 7x + 3 = 0 (22x + 11x2) + 7x + 3 = 0 Reorder the terms: 3 + 22x + 7x + 11x2 = 0 Combine like terms: 22x + 7x = 29x 3 + 29x + 11x2 = 0 Solving 3 + 29x + 11x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 0.2727272727 + 2.636363636x + x2 = 0 Move the constant term to the right: Add '-0.2727272727' to each side of the equation. 0.2727272727 + 2.636363636x + -0.2727272727 + x2 = 0 + -0.2727272727 Reorder the terms: 0.2727272727 + -0.2727272727 + 2.636363636x + x2 = 0 + -0.2727272727 Combine like terms: 0.2727272727 + -0.2727272727 = 0.0000000000 0.0000000000 + 2.636363636x + x2 = 0 + -0.2727272727 2.636363636x + x2 = 0 + -0.2727272727 Combine like terms: 0 + -0.2727272727 = -0.2727272727 2.636363636x + x2 = -0.2727272727 The x term is 2.636363636x. Take half its coefficient (1.318181818). Square it (1.737603305) and add it to both sides. Add '1.737603305' to each side of the equation. 2.636363636x + 1.737603305 + x2 = -0.2727272727 + 1.737603305 Reorder the terms: 1.737603305 + 2.636363636x + x2 = -0.2727272727 + 1.737603305 Combine like terms: -0.2727272727 + 1.737603305 = 1.4648760323 1.737603305 + 2.636363636x + x2 = 1.4648760323 Factor a perfect square on the left side: (x + 1.318181818)(x + 1.318181818) = 1.4648760323 Calculate the square root of the right side: 1.210320632 Break this problem into two subproblems by setting (x + 1.318181818) equal to 1.210320632 and -1.210320632.Subproblem 1
x + 1.318181818 = 1.210320632 Simplifying x + 1.318181818 = 1.210320632 Reorder the terms: 1.318181818 + x = 1.210320632 Solving 1.318181818 + x = 1.210320632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.318181818' to each side of the equation. 1.318181818 + -1.318181818 + x = 1.210320632 + -1.318181818 Combine like terms: 1.318181818 + -1.318181818 = 0.000000000 0.000000000 + x = 1.210320632 + -1.318181818 x = 1.210320632 + -1.318181818 Combine like terms: 1.210320632 + -1.318181818 = -0.107861186 x = -0.107861186 Simplifying x = -0.107861186Subproblem 2
x + 1.318181818 = -1.210320632 Simplifying x + 1.318181818 = -1.210320632 Reorder the terms: 1.318181818 + x = -1.210320632 Solving 1.318181818 + x = -1.210320632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.318181818' to each side of the equation. 1.318181818 + -1.318181818 + x = -1.210320632 + -1.318181818 Combine like terms: 1.318181818 + -1.318181818 = 0.000000000 0.000000000 + x = -1.210320632 + -1.318181818 x = -1.210320632 + -1.318181818 Combine like terms: -1.210320632 + -1.318181818 = -2.52850245 x = -2.52850245 Simplifying x = -2.52850245Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.107861186, -2.52850245}
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