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Simplifying 1 + -1(1 + -2x)(1 + 2x) = x + 2 Multiply (1 + -2x) * (1 + 2x) 1 + -1(1(1 + 2x) + -2x * (1 + 2x)) = x + 2 1 + -1((1 * 1 + 2x * 1) + -2x * (1 + 2x)) = x + 2 1 + -1((1 + 2x) + -2x * (1 + 2x)) = x + 2 1 + -1(1 + 2x + (1 * -2x + 2x * -2x)) = x + 2 1 + -1(1 + 2x + (-2x + -4x2)) = x + 2 Combine like terms: 2x + -2x = 0 1 + -1(1 + 0 + -4x2) = x + 2 1 + -1(1 + -4x2) = x + 2 1 + (1 * -1 + -4x2 * -1) = x + 2 1 + (-1 + 4x2) = x + 2 Combine like terms: 1 + -1 = 0 0 + 4x2 = x + 2 4x2 = x + 2 Reorder the terms: 4x2 = 2 + x Solving 4x2 = 2 + x Solving for variable 'x'. Reorder the terms: -2 + -1x + 4x2 = 2 + x + -2 + -1x Reorder the terms: -2 + -1x + 4x2 = 2 + -2 + x + -1x Combine like terms: 2 + -2 = 0 -2 + -1x + 4x2 = 0 + x + -1x -2 + -1x + 4x2 = x + -1x Combine like terms: x + -1x = 0 -2 + -1x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + -0.25x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.25x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.25x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.25x + x2 = 0 + 0.5 -0.25x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.25x + x2 = 0.5 The x term is -0.25x. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25x + 0.015625 + x2 = 0.5 + 0.015625 Reorder the terms: 0.015625 + -0.25x + x2 = 0.5 + 0.015625 Combine like terms: 0.5 + 0.015625 = 0.515625 0.015625 + -0.25x + x2 = 0.515625 Factor a perfect square on the left side: (x + -0.125)(x + -0.125) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (x + -0.125) equal to 0.718070331 and -0.718070331.Subproblem 1
x + -0.125 = 0.718070331 Simplifying x + -0.125 = 0.718070331 Reorder the terms: -0.125 + x = 0.718070331 Solving -0.125 + x = 0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = 0.718070331 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = 0.718070331 + 0.125 x = 0.718070331 + 0.125 Combine like terms: 0.718070331 + 0.125 = 0.843070331 x = 0.843070331 Simplifying x = 0.843070331Subproblem 2
x + -0.125 = -0.718070331 Simplifying x + -0.125 = -0.718070331 Reorder the terms: -0.125 + x = -0.718070331 Solving -0.125 + x = -0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = -0.718070331 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = -0.718070331 + 0.125 x = -0.718070331 + 0.125 Combine like terms: -0.718070331 + 0.125 = -0.593070331 x = -0.593070331 Simplifying x = -0.593070331Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.843070331, -0.593070331}
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