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Simplifying 10x(x + 14) + 50x + 20(3x + 5) = 1440 Reorder the terms: 10x(14 + x) + 50x + 20(3x + 5) = 1440 (14 * 10x + x * 10x) + 50x + 20(3x + 5) = 1440 (140x + 10x2) + 50x + 20(3x + 5) = 1440 Reorder the terms: 140x + 10x2 + 50x + 20(5 + 3x) = 1440 140x + 10x2 + 50x + (5 * 20 + 3x * 20) = 1440 140x + 10x2 + 50x + (100 + 60x) = 1440 Reorder the terms: 100 + 140x + 50x + 60x + 10x2 = 1440 Combine like terms: 140x + 50x = 190x 100 + 190x + 60x + 10x2 = 1440 Combine like terms: 190x + 60x = 250x 100 + 250x + 10x2 = 1440 Solving 100 + 250x + 10x2 = 1440 Solving for variable 'x'. Reorder the terms: 100 + -1440 + 250x + 10x2 = 1440 + -1440 Combine like terms: 100 + -1440 = -1340 -1340 + 250x + 10x2 = 1440 + -1440 Combine like terms: 1440 + -1440 = 0 -1340 + 250x + 10x2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(-134 + 25x + x2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-134 + 25x + x2)' equal to zero and attempt to solve: Simplifying -134 + 25x + x2 = 0 Solving -134 + 25x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '134' to each side of the equation. -134 + 25x + 134 + x2 = 0 + 134 Reorder the terms: -134 + 134 + 25x + x2 = 0 + 134 Combine like terms: -134 + 134 = 0 0 + 25x + x2 = 0 + 134 25x + x2 = 0 + 134 Combine like terms: 0 + 134 = 134 25x + x2 = 134 The x term is 25x. Take half its coefficient (12.5). Square it (156.25) and add it to both sides. Add '156.25' to each side of the equation. 25x + 156.25 + x2 = 134 + 156.25 Reorder the terms: 156.25 + 25x + x2 = 134 + 156.25 Combine like terms: 134 + 156.25 = 290.25 156.25 + 25x + x2 = 290.25 Factor a perfect square on the left side: (x + 12.5)(x + 12.5) = 290.25 Calculate the square root of the right side: 17.036725037 Break this problem into two subproblems by setting (x + 12.5) equal to 17.036725037 and -17.036725037.Subproblem 1
x + 12.5 = 17.036725037 Simplifying x + 12.5 = 17.036725037 Reorder the terms: 12.5 + x = 17.036725037 Solving 12.5 + x = 17.036725037 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + x = 17.036725037 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + x = 17.036725037 + -12.5 x = 17.036725037 + -12.5 Combine like terms: 17.036725037 + -12.5 = 4.536725037 x = 4.536725037 Simplifying x = 4.536725037Subproblem 2
x + 12.5 = -17.036725037 Simplifying x + 12.5 = -17.036725037 Reorder the terms: 12.5 + x = -17.036725037 Solving 12.5 + x = -17.036725037 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + x = -17.036725037 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + x = -17.036725037 + -12.5 x = -17.036725037 + -12.5 Combine like terms: -17.036725037 + -12.5 = -29.536725037 x = -29.536725037 Simplifying x = -29.536725037Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.536725037, -29.536725037}Solution
x = {4.536725037, -29.536725037}
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