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Simplifying (4x + 1)(x + -1) = 1 Reorder the terms: (1 + 4x)(x + -1) = 1 Reorder the terms: (1 + 4x)(-1 + x) = 1 Multiply (1 + 4x) * (-1 + x) (1(-1 + x) + 4x * (-1 + x)) = 1 ((-1 * 1 + x * 1) + 4x * (-1 + x)) = 1 ((-1 + 1x) + 4x * (-1 + x)) = 1 (-1 + 1x + (-1 * 4x + x * 4x)) = 1 (-1 + 1x + (-4x + 4x2)) = 1 Combine like terms: 1x + -4x = -3x (-1 + -3x + 4x2) = 1 Solving -1 + -3x + 4x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + -1 + -3x + 4x2 = 1 + -1 Combine like terms: -1 + -1 = -2 -2 + -3x + 4x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -2 + -3x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + -0.75x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.75x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.75x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.75x + x2 = 0 + 0.5 -0.75x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.75x + x2 = 0.5 The x term is -0.75x. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75x + 0.140625 + x2 = 0.5 + 0.140625 Reorder the terms: 0.140625 + -0.75x + x2 = 0.5 + 0.140625 Combine like terms: 0.5 + 0.140625 = 0.640625 0.140625 + -0.75x + x2 = 0.640625 Factor a perfect square on the left side: (x + -0.375)(x + -0.375) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (x + -0.375) equal to 0.80039053 and -0.80039053.Subproblem 1
x + -0.375 = 0.80039053 Simplifying x + -0.375 = 0.80039053 Reorder the terms: -0.375 + x = 0.80039053 Solving -0.375 + x = 0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 0.80039053 + 0.375 x = 0.80039053 + 0.375 Combine like terms: 0.80039053 + 0.375 = 1.17539053 x = 1.17539053 Simplifying x = 1.17539053Subproblem 2
x + -0.375 = -0.80039053 Simplifying x + -0.375 = -0.80039053 Reorder the terms: -0.375 + x = -0.80039053 Solving -0.375 + x = -0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -0.80039053 + 0.375 x = -0.80039053 + 0.375 Combine like terms: -0.80039053 + 0.375 = -0.42539053 x = -0.42539053 Simplifying x = -0.42539053Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.17539053, -0.42539053}
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