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Simplifying 19x + 2x2 = 1000 Solving 19x + 2x2 = 1000 Solving for variable 'x'. Reorder the terms: -1000 + 19x + 2x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 19x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -500 + 9.5x + x2 = 0 Move the constant term to the right: Add '500' to each side of the equation. -500 + 9.5x + 500 + x2 = 0 + 500 Reorder the terms: -500 + 500 + 9.5x + x2 = 0 + 500 Combine like terms: -500 + 500 = 0 0 + 9.5x + x2 = 0 + 500 9.5x + x2 = 0 + 500 Combine like terms: 0 + 500 = 500 9.5x + x2 = 500 The x term is 9.5x. Take half its coefficient (4.75). Square it (22.5625) and add it to both sides. Add '22.5625' to each side of the equation. 9.5x + 22.5625 + x2 = 500 + 22.5625 Reorder the terms: 22.5625 + 9.5x + x2 = 500 + 22.5625 Combine like terms: 500 + 22.5625 = 522.5625 22.5625 + 9.5x + x2 = 522.5625 Factor a perfect square on the left side: (x + 4.75)(x + 4.75) = 522.5625 Calculate the square root of the right side: 22.859625981 Break this problem into two subproblems by setting (x + 4.75) equal to 22.859625981 and -22.859625981.Subproblem 1
x + 4.75 = 22.859625981 Simplifying x + 4.75 = 22.859625981 Reorder the terms: 4.75 + x = 22.859625981 Solving 4.75 + x = 22.859625981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + x = 22.859625981 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + x = 22.859625981 + -4.75 x = 22.859625981 + -4.75 Combine like terms: 22.859625981 + -4.75 = 18.109625981 x = 18.109625981 Simplifying x = 18.109625981Subproblem 2
x + 4.75 = -22.859625981 Simplifying x + 4.75 = -22.859625981 Reorder the terms: 4.75 + x = -22.859625981 Solving 4.75 + x = -22.859625981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + x = -22.859625981 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + x = -22.859625981 + -4.75 x = -22.859625981 + -4.75 Combine like terms: -22.859625981 + -4.75 = -27.609625981 x = -27.609625981 Simplifying x = -27.609625981Solution
The solution to the problem is based on the solutions from the subproblems. x = {18.109625981, -27.609625981}
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