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Simplifying (4x + 5)(5x + 22) = 180 Reorder the terms: (5 + 4x)(5x + 22) = 180 Reorder the terms: (5 + 4x)(22 + 5x) = 180 Multiply (5 + 4x) * (22 + 5x) (5(22 + 5x) + 4x * (22 + 5x)) = 180 ((22 * 5 + 5x * 5) + 4x * (22 + 5x)) = 180 ((110 + 25x) + 4x * (22 + 5x)) = 180 (110 + 25x + (22 * 4x + 5x * 4x)) = 180 (110 + 25x + (88x + 20x2)) = 180 Combine like terms: 25x + 88x = 113x (110 + 113x + 20x2) = 180 Solving 110 + 113x + 20x2 = 180 Solving for variable 'x'. Reorder the terms: 110 + -180 + 113x + 20x2 = 180 + -180 Combine like terms: 110 + -180 = -70 -70 + 113x + 20x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -70 + 113x + 20x2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -3.5 + 5.65x + x2 = 0.0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 5.65x + 3.5 + x2 = 0.0 + 3.5 Reorder the terms: -3.5 + 3.5 + 5.65x + x2 = 0.0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 5.65x + x2 = 0.0 + 3.5 5.65x + x2 = 0.0 + 3.5 Combine like terms: 0.0 + 3.5 = 3.5 5.65x + x2 = 3.5 The x term is 5.65x. Take half its coefficient (2.825). Square it (7.980625) and add it to both sides. Add '7.980625' to each side of the equation. 5.65x + 7.980625 + x2 = 3.5 + 7.980625 Reorder the terms: 7.980625 + 5.65x + x2 = 3.5 + 7.980625 Combine like terms: 3.5 + 7.980625 = 11.480625 7.980625 + 5.65x + x2 = 11.480625 Factor a perfect square on the left side: (x + 2.825)(x + 2.825) = 11.480625 Calculate the square root of the right side: 3.388307099 Break this problem into two subproblems by setting (x + 2.825) equal to 3.388307099 and -3.388307099.Subproblem 1
x + 2.825 = 3.388307099 Simplifying x + 2.825 = 3.388307099 Reorder the terms: 2.825 + x = 3.388307099 Solving 2.825 + x = 3.388307099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.825' to each side of the equation. 2.825 + -2.825 + x = 3.388307099 + -2.825 Combine like terms: 2.825 + -2.825 = 0.000 0.000 + x = 3.388307099 + -2.825 x = 3.388307099 + -2.825 Combine like terms: 3.388307099 + -2.825 = 0.563307099 x = 0.563307099 Simplifying x = 0.563307099Subproblem 2
x + 2.825 = -3.388307099 Simplifying x + 2.825 = -3.388307099 Reorder the terms: 2.825 + x = -3.388307099 Solving 2.825 + x = -3.388307099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.825' to each side of the equation. 2.825 + -2.825 + x = -3.388307099 + -2.825 Combine like terms: 2.825 + -2.825 = 0.000 0.000 + x = -3.388307099 + -2.825 x = -3.388307099 + -2.825 Combine like terms: -3.388307099 + -2.825 = -6.213307099 x = -6.213307099 Simplifying x = -6.213307099Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.563307099, -6.213307099}
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