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Simplifying 0.5x2 + 1.5x + 2 = 2000 Reorder the terms: 2 + 1.5x + 0.5x2 = 2000 Solving 2 + 1.5x + 0.5x2 = 2000 Solving for variable 'x'. Reorder the terms: 2 + -2000 + 1.5x + 0.5x2 = 2000 + -2000 Combine like terms: 2 + -2000 = -1998 -1998 + 1.5x + 0.5x2 = 2000 + -2000 Combine like terms: 2000 + -2000 = 0 -1998 + 1.5x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -3996 + 3x + x2 = 0 Move the constant term to the right: Add '3996' to each side of the equation. -3996 + 3x + 3996 + x2 = 0 + 3996 Reorder the terms: -3996 + 3996 + 3x + x2 = 0 + 3996 Combine like terms: -3996 + 3996 = 0 0 + 3x + x2 = 0 + 3996 3x + x2 = 0 + 3996 Combine like terms: 0 + 3996 = 3996 3x + x2 = 3996 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 3996 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 3996 + 2.25 Combine like terms: 3996 + 2.25 = 3998.25 2.25 + 3x + x2 = 3998.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 3998.25 Calculate the square root of the right side: 63.231716725 Break this problem into two subproblems by setting (x + 1.5) equal to 63.231716725 and -63.231716725.Subproblem 1
x + 1.5 = 63.231716725 Simplifying x + 1.5 = 63.231716725 Reorder the terms: 1.5 + x = 63.231716725 Solving 1.5 + x = 63.231716725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 63.231716725 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 63.231716725 + -1.5 x = 63.231716725 + -1.5 Combine like terms: 63.231716725 + -1.5 = 61.731716725 x = 61.731716725 Simplifying x = 61.731716725Subproblem 2
x + 1.5 = -63.231716725 Simplifying x + 1.5 = -63.231716725 Reorder the terms: 1.5 + x = -63.231716725 Solving 1.5 + x = -63.231716725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -63.231716725 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -63.231716725 + -1.5 x = -63.231716725 + -1.5 Combine like terms: -63.231716725 + -1.5 = -64.731716725 x = -64.731716725 Simplifying x = -64.731716725Solution
The solution to the problem is based on the solutions from the subproblems. x = {61.731716725, -64.731716725}
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