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Simplifying (24 + 2x)(15 + 2x) = 540 Multiply (24 + 2x) * (15 + 2x) (24(15 + 2x) + 2x * (15 + 2x)) = 540 ((15 * 24 + 2x * 24) + 2x * (15 + 2x)) = 540 ((360 + 48x) + 2x * (15 + 2x)) = 540 (360 + 48x + (15 * 2x + 2x * 2x)) = 540 (360 + 48x + (30x + 4x2)) = 540 Combine like terms: 48x + 30x = 78x (360 + 78x + 4x2) = 540 Solving 360 + 78x + 4x2 = 540 Solving for variable 'x'. Reorder the terms: 360 + -540 + 78x + 4x2 = 540 + -540 Combine like terms: 360 + -540 = -180 -180 + 78x + 4x2 = 540 + -540 Combine like terms: 540 + -540 = 0 -180 + 78x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-90 + 39x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-90 + 39x + 2x2)' equal to zero and attempt to solve: Simplifying -90 + 39x + 2x2 = 0 Solving -90 + 39x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -45 + 19.5x + x2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 19.5x + 45 + x2 = 0 + 45 Reorder the terms: -45 + 45 + 19.5x + x2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 19.5x + x2 = 0 + 45 19.5x + x2 = 0 + 45 Combine like terms: 0 + 45 = 45 19.5x + x2 = 45 The x term is 19.5x. Take half its coefficient (9.75). Square it (95.0625) and add it to both sides. Add '95.0625' to each side of the equation. 19.5x + 95.0625 + x2 = 45 + 95.0625 Reorder the terms: 95.0625 + 19.5x + x2 = 45 + 95.0625 Combine like terms: 45 + 95.0625 = 140.0625 95.0625 + 19.5x + x2 = 140.0625 Factor a perfect square on the left side: (x + 9.75)(x + 9.75) = 140.0625 Calculate the square root of the right side: 11.834800379 Break this problem into two subproblems by setting (x + 9.75) equal to 11.834800379 and -11.834800379.Subproblem 1
x + 9.75 = 11.834800379 Simplifying x + 9.75 = 11.834800379 Reorder the terms: 9.75 + x = 11.834800379 Solving 9.75 + x = 11.834800379 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9.75' to each side of the equation. 9.75 + -9.75 + x = 11.834800379 + -9.75 Combine like terms: 9.75 + -9.75 = 0.00 0.00 + x = 11.834800379 + -9.75 x = 11.834800379 + -9.75 Combine like terms: 11.834800379 + -9.75 = 2.084800379 x = 2.084800379 Simplifying x = 2.084800379Subproblem 2
x + 9.75 = -11.834800379 Simplifying x + 9.75 = -11.834800379 Reorder the terms: 9.75 + x = -11.834800379 Solving 9.75 + x = -11.834800379 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9.75' to each side of the equation. 9.75 + -9.75 + x = -11.834800379 + -9.75 Combine like terms: 9.75 + -9.75 = 0.00 0.00 + x = -11.834800379 + -9.75 x = -11.834800379 + -9.75 Combine like terms: -11.834800379 + -9.75 = -21.584800379 x = -21.584800379 Simplifying x = -21.584800379Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.084800379, -21.584800379}Solution
x = {2.084800379, -21.584800379}
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