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Simplifying (2x + 60)(2x + 30) = 82503 Reorder the terms: (60 + 2x)(2x + 30) = 82503 Reorder the terms: (60 + 2x)(30 + 2x) = 82503 Multiply (60 + 2x) * (30 + 2x) (60(30 + 2x) + 2x * (30 + 2x)) = 82503 ((30 * 60 + 2x * 60) + 2x * (30 + 2x)) = 82503 ((1800 + 120x) + 2x * (30 + 2x)) = 82503 (1800 + 120x + (30 * 2x + 2x * 2x)) = 82503 (1800 + 120x + (60x + 4x2)) = 82503 Combine like terms: 120x + 60x = 180x (1800 + 180x + 4x2) = 82503 Solving 1800 + 180x + 4x2 = 82503 Solving for variable 'x'. Reorder the terms: 1800 + -82503 + 180x + 4x2 = 82503 + -82503 Combine like terms: 1800 + -82503 = -80703 -80703 + 180x + 4x2 = 82503 + -82503 Combine like terms: 82503 + -82503 = 0 -80703 + 180x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -20175.75 + 45x + x2 = 0 Move the constant term to the right: Add '20175.75' to each side of the equation. -20175.75 + 45x + 20175.75 + x2 = 0 + 20175.75 Reorder the terms: -20175.75 + 20175.75 + 45x + x2 = 0 + 20175.75 Combine like terms: -20175.75 + 20175.75 = 0.00 0.00 + 45x + x2 = 0 + 20175.75 45x + x2 = 0 + 20175.75 Combine like terms: 0 + 20175.75 = 20175.75 45x + x2 = 20175.75 The x term is 45x. Take half its coefficient (22.5). Square it (506.25) and add it to both sides. Add '506.25' to each side of the equation. 45x + 506.25 + x2 = 20175.75 + 506.25 Reorder the terms: 506.25 + 45x + x2 = 20175.75 + 506.25 Combine like terms: 20175.75 + 506.25 = 20682 506.25 + 45x + x2 = 20682 Factor a perfect square on the left side: (x + 22.5)(x + 22.5) = 20682 Calculate the square root of the right side: 143.81237777 Break this problem into two subproblems by setting (x + 22.5) equal to 143.81237777 and -143.81237777.Subproblem 1
x + 22.5 = 143.81237777 Simplifying x + 22.5 = 143.81237777 Reorder the terms: 22.5 + x = 143.81237777 Solving 22.5 + x = 143.81237777 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = 143.81237777 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = 143.81237777 + -22.5 x = 143.81237777 + -22.5 Combine like terms: 143.81237777 + -22.5 = 121.31237777 x = 121.31237777 Simplifying x = 121.31237777Subproblem 2
x + 22.5 = -143.81237777 Simplifying x + 22.5 = -143.81237777 Reorder the terms: 22.5 + x = -143.81237777 Solving 22.5 + x = -143.81237777 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = -143.81237777 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = -143.81237777 + -22.5 x = -143.81237777 + -22.5 Combine like terms: -143.81237777 + -22.5 = -166.31237777 x = -166.31237777 Simplifying x = -166.31237777Solution
The solution to the problem is based on the solutions from the subproblems. x = {121.31237777, -166.31237777}
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