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Simplifying 10x2 + -5x + 11 = -1x2 + x + 83 Reorder the terms: 11 + -5x + 10x2 = -1x2 + x + 83 Reorder the terms: 11 + -5x + 10x2 = 83 + x + -1x2 Solving 11 + -5x + 10x2 = 83 + x + -1x2 Solving for variable 'x'. Reorder the terms: 11 + -83 + -5x + -1x + 10x2 + x2 = 83 + x + -1x2 + -83 + -1x + x2 Combine like terms: 11 + -83 = -72 -72 + -5x + -1x + 10x2 + x2 = 83 + x + -1x2 + -83 + -1x + x2 Combine like terms: -5x + -1x = -6x -72 + -6x + 10x2 + x2 = 83 + x + -1x2 + -83 + -1x + x2 Combine like terms: 10x2 + x2 = 11x2 -72 + -6x + 11x2 = 83 + x + -1x2 + -83 + -1x + x2 Reorder the terms: -72 + -6x + 11x2 = 83 + -83 + x + -1x + -1x2 + x2 Combine like terms: 83 + -83 = 0 -72 + -6x + 11x2 = 0 + x + -1x + -1x2 + x2 -72 + -6x + 11x2 = x + -1x + -1x2 + x2 Combine like terms: x + -1x = 0 -72 + -6x + 11x2 = 0 + -1x2 + x2 -72 + -6x + 11x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 -72 + -6x + 11x2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -6.545454545 + -0.5454545455x + x2 = 0 Move the constant term to the right: Add '6.545454545' to each side of the equation. -6.545454545 + -0.5454545455x + 6.545454545 + x2 = 0 + 6.545454545 Reorder the terms: -6.545454545 + 6.545454545 + -0.5454545455x + x2 = 0 + 6.545454545 Combine like terms: -6.545454545 + 6.545454545 = 0.000000000 0.000000000 + -0.5454545455x + x2 = 0 + 6.545454545 -0.5454545455x + x2 = 0 + 6.545454545 Combine like terms: 0 + 6.545454545 = 6.545454545 -0.5454545455x + x2 = 6.545454545 The x term is -0.5454545455x. Take half its coefficient (-0.2727272728). Square it (0.07438016533) and add it to both sides. Add '0.07438016533' to each side of the equation. -0.5454545455x + 0.07438016533 + x2 = 6.545454545 + 0.07438016533 Reorder the terms: 0.07438016533 + -0.5454545455x + x2 = 6.545454545 + 0.07438016533 Combine like terms: 6.545454545 + 0.07438016533 = 6.61983471033 0.07438016533 + -0.5454545455x + x2 = 6.61983471033 Factor a perfect square on the left side: (x + -0.2727272728)(x + -0.2727272728) = 6.61983471033 Calculate the square root of the right side: 2.572903945 Break this problem into two subproblems by setting (x + -0.2727272728) equal to 2.572903945 and -2.572903945.Subproblem 1
x + -0.2727272728 = 2.572903945 Simplifying x + -0.2727272728 = 2.572903945 Reorder the terms: -0.2727272728 + x = 2.572903945 Solving -0.2727272728 + x = 2.572903945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2727272728' to each side of the equation. -0.2727272728 + 0.2727272728 + x = 2.572903945 + 0.2727272728 Combine like terms: -0.2727272728 + 0.2727272728 = 0.0000000000 0.0000000000 + x = 2.572903945 + 0.2727272728 x = 2.572903945 + 0.2727272728 Combine like terms: 2.572903945 + 0.2727272728 = 2.8456312178 x = 2.8456312178 Simplifying x = 2.8456312178Subproblem 2
x + -0.2727272728 = -2.572903945 Simplifying x + -0.2727272728 = -2.572903945 Reorder the terms: -0.2727272728 + x = -2.572903945 Solving -0.2727272728 + x = -2.572903945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2727272728' to each side of the equation. -0.2727272728 + 0.2727272728 + x = -2.572903945 + 0.2727272728 Combine like terms: -0.2727272728 + 0.2727272728 = 0.0000000000 0.0000000000 + x = -2.572903945 + 0.2727272728 x = -2.572903945 + 0.2727272728 Combine like terms: -2.572903945 + 0.2727272728 = -2.3001766722 x = -2.3001766722 Simplifying x = -2.3001766722Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.8456312178, -2.3001766722}
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