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Simplifying x(x + 1) = 2147483647 Reorder the terms: x(1 + x) = 2147483647 (1 * x + x * x) = 2147483647 (1x + x2) = 2147483647 Solving 1x + x2 = 2147483647 Solving for variable 'x'. Reorder the terms: -2147483647 + 1x + x2 = 2147483647 + -2147483647 Combine like terms: 2147483647 + -2147483647 = 0 -2147483647 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '2147483647' to each side of the equation. -2147483647 + 1x + 2147483647 + x2 = 0 + 2147483647 Reorder the terms: -2147483647 + 2147483647 + 1x + x2 = 0 + 2147483647 Combine like terms: -2147483647 + 2147483647 = 0 0 + 1x + x2 = 0 + 2147483647 1x + x2 = 0 + 2147483647 Combine like terms: 0 + 2147483647 = 2147483647 1x + x2 = 2147483647 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. x + 0.25 + x2 = 2147483647 + 0.25 Reorder the terms: 0.25 + x + x2 = 2147483647 + 0.25 Combine like terms: 2147483647 + 0.25 = 2147483647.25 0.25 + x + x2 = 2147483647.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 2147483647.25 Calculate the square root of the right side: 46340.950003749 Break this problem into two subproblems by setting (x + 0.5) equal to 46340.950003749 and -46340.950003749.Subproblem 1
x + 0.5 = 46340.950003749 Simplifying x + 0.5 = 46340.950003749 Reorder the terms: 0.5 + x = 46340.950003749 Solving 0.5 + x = 46340.950003749 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 = 46340.950003749 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 46340.950003749 + -0.5 x = 46340.950003749 + -0.5 Combine like terms: 46340.950003749 + -0.5 = 46340.450003749 x = 46340.450003749 Simplifying x = 46340.450003749Subproblem 2
x + 0.5 = -46340.950003749 Simplifying x + 0.5 = -46340.950003749 Reorder the terms: 0.5 + x = -46340.950003749 Solving 0.5 + x = -46340.950003749 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 = -46340.950003749 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -46340.950003749 + -0.5 x = -46340.950003749 + -0.5 Combine like terms: -46340.950003749 + -0.5 = -46341.450003749 x = -46341.450003749 Simplifying x = -46341.450003749Solution
The solution to the problem is based on the solutions from the subproblems. x = {46340.450003749, -46341.450003749}
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