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Simplifying (x + 1)(x + -1) + -1(x + -1) = 2012 Reorder the terms: (1 + x)(x + -1) + -1(x + -1) = 2012 Reorder the terms: (1 + x)(-1 + x) + -1(x + -1) = 2012 Multiply (1 + x) * (-1 + x) (1(-1 + x) + x(-1 + x)) + -1(x + -1) = 2012 ((-1 * 1 + x * 1) + x(-1 + x)) + -1(x + -1) = 2012 ((-1 + 1x) + x(-1 + x)) + -1(x + -1) = 2012 (-1 + 1x + (-1 * x + x * x)) + -1(x + -1) = 2012 (-1 + 1x + (-1x + x2)) + -1(x + -1) = 2012 Combine like terms: 1x + -1x = 0 (-1 + 0 + x2) + -1(x + -1) = 2012 (-1 + x2) + -1(x + -1) = 2012 Reorder the terms: -1 + x2 + -1(-1 + x) = 2012 -1 + x2 + (-1 * -1 + x * -1) = 2012 -1 + x2 + (1 + -1x) = 2012 Reorder the terms: -1 + 1 + -1x + x2 = 2012 Combine like terms: -1 + 1 = 0 0 + -1x + x2 = 2012 -1x + x2 = 2012 Solving -1x + x2 = 2012 Solving for variable 'x'. Reorder the terms: -2012 + -1x + x2 = 2012 + -2012 Combine like terms: 2012 + -2012 = 0 -2012 + -1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '2012' to each side of the equation. -2012 + -1x + 2012 + x2 = 0 + 2012 Reorder the terms: -2012 + 2012 + -1x + x2 = 0 + 2012 Combine like terms: -2012 + 2012 = 0 0 + -1x + x2 = 0 + 2012 -1x + x2 = 0 + 2012 Combine like terms: 0 + 2012 = 2012 -1x + x2 = 2012 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 = 2012 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 2012 + 0.25 Combine like terms: 2012 + 0.25 = 2012.25 0.25 + -1x + x2 = 2012.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 2012.25 Calculate the square root of the right side: 44.858109635 Break this problem into two subproblems by setting (x + -0.5) equal to 44.858109635 and -44.858109635.Subproblem 1
x + -0.5 = 44.858109635 Simplifying x + -0.5 = 44.858109635 Reorder the terms: -0.5 + x = 44.858109635 Solving -0.5 + x = 44.858109635 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 = 44.858109635 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 44.858109635 + 0.5 x = 44.858109635 + 0.5 Combine like terms: 44.858109635 + 0.5 = 45.358109635 x = 45.358109635 Simplifying x = 45.358109635Subproblem 2
x + -0.5 = -44.858109635 Simplifying x + -0.5 = -44.858109635 Reorder the terms: -0.5 + x = -44.858109635 Solving -0.5 + x = -44.858109635 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 = -44.858109635 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -44.858109635 + 0.5 x = -44.858109635 + 0.5 Combine like terms: -44.858109635 + 0.5 = -44.358109635 x = -44.358109635 Simplifying x = -44.358109635Solution
The solution to the problem is based on the solutions from the subproblems. x = {45.358109635, -44.358109635}
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