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Simplifying 1000 = x(x + -2) Reorder the terms: 1000 = x(-2 + x) 1000 = (-2 * x + x * x) 1000 = (-2x + x2) Solving 1000 = -2x + x2 Solving for variable 'x'. Reorder the terms: 1000 + 2x + -1x2 = -2x + 2x + x2 + -1x2 Combine like terms: -2x + 2x = 0 1000 + 2x + -1x2 = 0 + x2 + -1x2 1000 + 2x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 1000 + 2x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1000 + -2x + x2 = 0 Move the constant term to the right: Add '1000' to each side of the equation. -1000 + -2x + 1000 + x2 = 0 + 1000 Reorder the terms: -1000 + 1000 + -2x + x2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + -2x + x2 = 0 + 1000 -2x + x2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 -2x + x2 = 1000 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = 1000 + 1 Reorder the terms: 1 + -2x + x2 = 1000 + 1 Combine like terms: 1000 + 1 = 1001 1 + -2x + x2 = 1001 Factor a perfect square on the left side: (x + -1)(x + -1) = 1001 Calculate the square root of the right side: 31.638584039 Break this problem into two subproblems by setting (x + -1) equal to 31.638584039 and -31.638584039.Subproblem 1
x + -1 = 31.638584039 Simplifying x + -1 = 31.638584039 Reorder the terms: -1 + x = 31.638584039 Solving -1 + x = 31.638584039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 31.638584039 + 1 Combine like terms: -1 + 1 = 0 0 + x = 31.638584039 + 1 x = 31.638584039 + 1 Combine like terms: 31.638584039 + 1 = 32.638584039 x = 32.638584039 Simplifying x = 32.638584039Subproblem 2
x + -1 = -31.638584039 Simplifying x + -1 = -31.638584039 Reorder the terms: -1 + x = -31.638584039 Solving -1 + x = -31.638584039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -31.638584039 + 1 Combine like terms: -1 + 1 = 0 0 + x = -31.638584039 + 1 x = -31.638584039 + 1 Combine like terms: -31.638584039 + 1 = -30.638584039 x = -30.638584039 Simplifying x = -30.638584039Solution
The solution to the problem is based on the solutions from the subproblems. x = {32.638584039, -30.638584039}
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