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Simplifying 1 + x + x2 = 108 Solving 1 + x + x2 = 108 Solving for variable 'x'. Reorder the terms: 1 + -108 + x + x2 = 108 + -108 Combine like terms: 1 + -108 = -107 -107 + x + x2 = 108 + -108 Combine like terms: 108 + -108 = 0 -107 + x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '107' to each side of the equation. -107 + x + 107 + x2 = 0 + 107 Reorder the terms: -107 + 107 + x + x2 = 0 + 107 Combine like terms: -107 + 107 = 0 0 + x + x2 = 0 + 107 x + x2 = 0 + 107 Combine like terms: 0 + 107 = 107 x + x2 = 107 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. x + 0.25 + x2 = 107 + 0.25 Reorder the terms: 0.25 + x + x2 = 107 + 0.25 Combine like terms: 107 + 0.25 = 107.25 0.25 + x + x2 = 107.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 107.25 Calculate the square root of the right side: 10.356157589 Break this problem into two subproblems by setting (x + 0.5) equal to 10.356157589 and -10.356157589.Subproblem 1
x + 0.5 = 10.356157589 Simplifying x + 0.5 = 10.356157589 Reorder the terms: 0.5 + x = 10.356157589 Solving 0.5 + x = 10.356157589 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 = 10.356157589 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 10.356157589 + -0.5 x = 10.356157589 + -0.5 Combine like terms: 10.356157589 + -0.5 = 9.856157589 x = 9.856157589 Simplifying x = 9.856157589Subproblem 2
x + 0.5 = -10.356157589 Simplifying x + 0.5 = -10.356157589 Reorder the terms: 0.5 + x = -10.356157589 Solving 0.5 + x = -10.356157589 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 = -10.356157589 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -10.356157589 + -0.5 x = -10.356157589 + -0.5 Combine like terms: -10.356157589 + -0.5 = -10.856157589 x = -10.856157589 Simplifying x = -10.856157589Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.856157589, -10.856157589}
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