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Simplifying 8x + 2x(2x + 11) = x + 2000 Reorder the terms: 8x + 2x(11 + 2x) = x + 2000 8x + (11 * 2x + 2x * 2x) = x + 2000 8x + (22x + 4x2) = x + 2000 Combine like terms: 8x + 22x = 30x 30x + 4x2 = x + 2000 Reorder the terms: 30x + 4x2 = 2000 + x Solving 30x + 4x2 = 2000 + x Solving for variable 'x'. Reorder the terms: -2000 + 30x + -1x + 4x2 = 2000 + x + -2000 + -1x Combine like terms: 30x + -1x = 29x -2000 + 29x + 4x2 = 2000 + x + -2000 + -1x Reorder the terms: -2000 + 29x + 4x2 = 2000 + -2000 + x + -1x Combine like terms: 2000 + -2000 = 0 -2000 + 29x + 4x2 = 0 + x + -1x -2000 + 29x + 4x2 = x + -1x Combine like terms: x + -1x = 0 -2000 + 29x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -500 + 7.25x + x2 = 0 Move the constant term to the right: Add '500' to each side of the equation. -500 + 7.25x + 500 + x2 = 0 + 500 Reorder the terms: -500 + 500 + 7.25x + x2 = 0 + 500 Combine like terms: -500 + 500 = 0 0 + 7.25x + x2 = 0 + 500 7.25x + x2 = 0 + 500 Combine like terms: 0 + 500 = 500 7.25x + x2 = 500 The x term is 7.25x. Take half its coefficient (3.625). Square it (13.140625) and add it to both sides. Add '13.140625' to each side of the equation. 7.25x + 13.140625 + x2 = 500 + 13.140625 Reorder the terms: 13.140625 + 7.25x + x2 = 500 + 13.140625 Combine like terms: 500 + 13.140625 = 513.140625 13.140625 + 7.25x + x2 = 513.140625 Factor a perfect square on the left side: (x + 3.625)(x + 3.625) = 513.140625 Calculate the square root of the right side: 22.652607466 Break this problem into two subproblems by setting (x + 3.625) equal to 22.652607466 and -22.652607466.Subproblem 1
x + 3.625 = 22.652607466 Simplifying x + 3.625 = 22.652607466 Reorder the terms: 3.625 + x = 22.652607466 Solving 3.625 + x = 22.652607466 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.625' to each side of the equation. 3.625 + -3.625 + x = 22.652607466 + -3.625 Combine like terms: 3.625 + -3.625 = 0.000 0.000 + x = 22.652607466 + -3.625 x = 22.652607466 + -3.625 Combine like terms: 22.652607466 + -3.625 = 19.027607466 x = 19.027607466 Simplifying x = 19.027607466Subproblem 2
x + 3.625 = -22.652607466 Simplifying x + 3.625 = -22.652607466 Reorder the terms: 3.625 + x = -22.652607466 Solving 3.625 + x = -22.652607466 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.625' to each side of the equation. 3.625 + -3.625 + x = -22.652607466 + -3.625 Combine like terms: 3.625 + -3.625 = 0.000 0.000 + x = -22.652607466 + -3.625 x = -22.652607466 + -3.625 Combine like terms: -22.652607466 + -3.625 = -26.277607466 x = -26.277607466 Simplifying x = -26.277607466Solution
The solution to the problem is based on the solutions from the subproblems. x = {19.027607466, -26.277607466}
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