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Simplifying (5 + 0.5x)(300 + -20x) = 1562.5 Multiply (5 + 0.5x) * (300 + -20x) (5(300 + -20x) + 0.5x * (300 + -20x)) = 1562.5 ((300 * 5 + -20x * 5) + 0.5x * (300 + -20x)) = 1562.5 ((1500 + -100x) + 0.5x * (300 + -20x)) = 1562.5 (1500 + -100x + (300 * 0.5x + -20x * 0.5x)) = 1562.5 (1500 + -100x + (150x + -10x2)) = 1562.5 Combine like terms: -100x + 150x = 50x (1500 + 50x + -10x2) = 1562.5 Solving 1500 + 50x + -10x2 = 1562.5 Solving for variable 'x'. Reorder the terms: 1500 + -1562.5 + 50x + -10x2 = 1562.5 + -1562.5 Combine like terms: 1500 + -1562.5 = -62.5 -62.5 + 50x + -10x2 = 1562.5 + -1562.5 Combine like terms: 1562.5 + -1562.5 = 0.0 -62.5 + 50x + -10x2 = 0.0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. 6.25 + -5x + x2 = 0.00 Move the constant term to the right: Add '-6.25' to each side of the equation. 6.25 + -5x + -6.25 + x2 = 0.00 + -6.25 Reorder the terms: 6.25 + -6.25 + -5x + x2 = 0.00 + -6.25 Combine like terms: 6.25 + -6.25 = 0.00 0.00 + -5x + x2 = 0.00 + -6.25 -5x + x2 = 0.00 + -6.25 Combine like terms: 0.00 + -6.25 = -6.25 -5x + x2 = -6.25 The x term is -5x. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5x + 6.25 + x2 = -6.25 + 6.25 Reorder the terms: 6.25 + -5x + x2 = -6.25 + 6.25 Combine like terms: -6.25 + 6.25 = 0.00 6.25 + -5x + x2 = 0.00 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 0.00 Calculate the square root of the right side: 0.00 Break this problem into two subproblems by setting (x + -2.5) equal to 0.00 and 0.00.Subproblem 1
x + -2.5 = 0.00 Simplifying x + -2.5 = 0.00 Reorder the terms: -2.5 + x = 0.00 Solving -2.5 + x = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 0.00 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 0.00 + 2.5 x = 0.00 + 2.5 Combine like terms: 0.00 + 2.5 = 2.5 x = 2.5 Simplifying x = 2.5Subproblem 2
x + -2.5 = 0.00 Simplifying x + -2.5 = 0.00 Reorder the terms: -2.5 + x = 0.00 Solving -2.5 + x = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 0.00 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 0.00 + 2.5 x = 0.00 + 2.5 Combine like terms: 0.00 + 2.5 = 2.5 x = 2.5 Simplifying x = 2.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.5, 2.5}
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