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Simplifying x(x + 1) = x + x + 1 Reorder the terms: x(1 + x) = x + x + 1 (1 * x + x * x) = x + x + 1 (1x + x2) = x + x + 1 Reorder the terms: 1x + x2 = 1 + x + x Combine like terms: x + x = 2x 1x + x2 = 1 + 2x Solving 1x + x2 = 1 + 2x Solving for variable 'x'. Reorder the terms: -1 + 1x + -2x + x2 = 1 + 2x + -1 + -2x Combine like terms: 1x + -2x = -1x -1 + -1x + x2 = 1 + 2x + -1 + -2x Reorder the terms: -1 + -1x + x2 = 1 + -1 + 2x + -2x Combine like terms: 1 + -1 = 0 -1 + -1x + x2 = 0 + 2x + -2x -1 + -1x + x2 = 2x + -2x Combine like terms: 2x + -2x = 0 -1 + -1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -1x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + -1x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1x + x2 = 0 + 1 -1x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1x + x2 = 1 The x term is -1x. 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. -1x + 0.25 + x2 = 1 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1x + x2 = 1.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + -0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + -0.5 = 1.118033989 Simplifying x + -0.5 = 1.118033989 Reorder the terms: -0.5 + x = 1.118033989 Solving -0.5 + x = 1.118033989 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 = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.118033989 + 0.5 x = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 x = 1.618033989 Simplifying x = 1.618033989Subproblem 2
x + -0.5 = -1.118033989 Simplifying x + -0.5 = -1.118033989 Reorder the terms: -0.5 + x = -1.118033989 Solving -0.5 + x = -1.118033989 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 = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.118033989 + 0.5 x = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 x = -0.618033989 Simplifying x = -0.618033989Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.618033989, -0.618033989}
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