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Simplifying 110x2 + x + 11 = 0 Reorder the terms: 11 + x + 110x2 = 0 Solving 11 + x + 110x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 110 the coefficient of the squared term: Divide each side by '110'. 0.1 + 0.009090909091x + x2 = 0 Move the constant term to the right: Add '-0.1' to each side of the equation. 0.1 + 0.009090909091x + -0.1 + x2 = 0 + -0.1 Reorder the terms: 0.1 + -0.1 + 0.009090909091x + x2 = 0 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + 0.009090909091x + x2 = 0 + -0.1 0.009090909091x + x2 = 0 + -0.1 Combine like terms: 0 + -0.1 = -0.1 0.009090909091x + x2 = -0.1 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.009090909091x + 0.25 + x2 = -0.1 + 0.25 Reorder the terms: 0.25 + 0.009090909091x + x2 = -0.1 + 0.25 Combine like terms: -0.1 + 0.25 = 0.15 0.25 + 0.009090909091x + x2 = 0.15 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.15 Calculate the square root of the right side: 0.387298335 Break this problem into two subproblems by setting (x + 0.5) equal to 0.387298335 and -0.387298335.Subproblem 1
x + 0.5 = 0.387298335 Simplifying x + 0.5 = 0.387298335 Reorder the terms: 0.5 + x = 0.387298335 Solving 0.5 + x = 0.387298335 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.387298335 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.387298335 + -0.5 x = 0.387298335 + -0.5 Combine like terms: 0.387298335 + -0.5 = -0.112701665 x = -0.112701665 Simplifying x = -0.112701665Subproblem 2
x + 0.5 = -0.387298335 Simplifying x + 0.5 = -0.387298335 Reorder the terms: 0.5 + x = -0.387298335 Solving 0.5 + x = -0.387298335 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.387298335 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.387298335 + -0.5 x = -0.387298335 + -0.5 Combine like terms: -0.387298335 + -0.5 = -0.887298335 x = -0.887298335 Simplifying x = -0.887298335Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.112701665, -0.887298335}
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