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Simplifying 213x + 2x2 = 2000 Solving 213x + 2x2 = 2000 Solving for variable 'x'. Reorder the terms: -2000 + 213x + 2x2 = 2000 + -2000 Combine like terms: 2000 + -2000 = 0 -2000 + 213x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1000 + 106.5x + x2 = 0 Move the constant term to the right: Add '1000' to each side of the equation. -1000 + 106.5x + 1000 + x2 = 0 + 1000 Reorder the terms: -1000 + 1000 + 106.5x + x2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + 106.5x + x2 = 0 + 1000 106.5x + x2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 106.5x + x2 = 1000 The x term is 106.5x. Take half its coefficient (53.25). Square it (2835.5625) and add it to both sides. Add '2835.5625' to each side of the equation. 106.5x + 2835.5625 + x2 = 1000 + 2835.5625 Reorder the terms: 2835.5625 + 106.5x + x2 = 1000 + 2835.5625 Combine like terms: 1000 + 2835.5625 = 3835.5625 2835.5625 + 106.5x + x2 = 3835.5625 Factor a perfect square on the left side: (x + 53.25)(x + 53.25) = 3835.5625 Calculate the square root of the right side: 61.931918265 Break this problem into two subproblems by setting (x + 53.25) equal to 61.931918265 and -61.931918265.Subproblem 1
x + 53.25 = 61.931918265 Simplifying x + 53.25 = 61.931918265 Reorder the terms: 53.25 + x = 61.931918265 Solving 53.25 + x = 61.931918265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-53.25' to each side of the equation. 53.25 + -53.25 + x = 61.931918265 + -53.25 Combine like terms: 53.25 + -53.25 = 0.00 0.00 + x = 61.931918265 + -53.25 x = 61.931918265 + -53.25 Combine like terms: 61.931918265 + -53.25 = 8.681918265 x = 8.681918265 Simplifying x = 8.681918265Subproblem 2
x + 53.25 = -61.931918265 Simplifying x + 53.25 = -61.931918265 Reorder the terms: 53.25 + x = -61.931918265 Solving 53.25 + x = -61.931918265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-53.25' to each side of the equation. 53.25 + -53.25 + x = -61.931918265 + -53.25 Combine like terms: 53.25 + -53.25 = 0.00 0.00 + x = -61.931918265 + -53.25 x = -61.931918265 + -53.25 Combine like terms: -61.931918265 + -53.25 = -115.181918265 x = -115.181918265 Simplifying x = -115.181918265Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.681918265, -115.181918265}
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