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Simplifying (2x + 5)(x + 12) * 10 = 5000 Reorder the terms: (5 + 2x)(x + 12) * 10 = 5000 Reorder the terms: (5 + 2x)(12 + x) * 10 = 5000 Reorder the terms for easier multiplication: 10(5 + 2x)(12 + x) = 5000 Multiply (5 + 2x) * (12 + x) 10(5(12 + x) + 2x * (12 + x)) = 5000 10((12 * 5 + x * 5) + 2x * (12 + x)) = 5000 10((60 + 5x) + 2x * (12 + x)) = 5000 10(60 + 5x + (12 * 2x + x * 2x)) = 5000 10(60 + 5x + (24x + 2x2)) = 5000 Combine like terms: 5x + 24x = 29x 10(60 + 29x + 2x2) = 5000 (60 * 10 + 29x * 10 + 2x2 * 10) = 5000 (600 + 290x + 20x2) = 5000 Solving 600 + 290x + 20x2 = 5000 Solving for variable 'x'. Reorder the terms: 600 + -5000 + 290x + 20x2 = 5000 + -5000 Combine like terms: 600 + -5000 = -4400 -4400 + 290x + 20x2 = 5000 + -5000 Combine like terms: 5000 + -5000 = 0 -4400 + 290x + 20x2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(-440 + 29x + 2x2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-440 + 29x + 2x2)' equal to zero and attempt to solve: Simplifying -440 + 29x + 2x2 = 0 Solving -440 + 29x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -220 + 14.5x + x2 = 0 Move the constant term to the right: Add '220' to each side of the equation. -220 + 14.5x + 220 + x2 = 0 + 220 Reorder the terms: -220 + 220 + 14.5x + x2 = 0 + 220 Combine like terms: -220 + 220 = 0 0 + 14.5x + x2 = 0 + 220 14.5x + x2 = 0 + 220 Combine like terms: 0 + 220 = 220 14.5x + x2 = 220 The x term is 14.5x. Take half its coefficient (7.25). Square it (52.5625) and add it to both sides. Add '52.5625' to each side of the equation. 14.5x + 52.5625 + x2 = 220 + 52.5625 Reorder the terms: 52.5625 + 14.5x + x2 = 220 + 52.5625 Combine like terms: 220 + 52.5625 = 272.5625 52.5625 + 14.5x + x2 = 272.5625 Factor a perfect square on the left side: (x + 7.25)(x + 7.25) = 272.5625 Calculate the square root of the right side: 16.509466981 Break this problem into two subproblems by setting (x + 7.25) equal to 16.509466981 and -16.509466981.Subproblem 1
x + 7.25 = 16.509466981 Simplifying x + 7.25 = 16.509466981 Reorder the terms: 7.25 + x = 16.509466981 Solving 7.25 + x = 16.509466981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.25' to each side of the equation. 7.25 + -7.25 + x = 16.509466981 + -7.25 Combine like terms: 7.25 + -7.25 = 0.00 0.00 + x = 16.509466981 + -7.25 x = 16.509466981 + -7.25 Combine like terms: 16.509466981 + -7.25 = 9.259466981 x = 9.259466981 Simplifying x = 9.259466981Subproblem 2
x + 7.25 = -16.509466981 Simplifying x + 7.25 = -16.509466981 Reorder the terms: 7.25 + x = -16.509466981 Solving 7.25 + x = -16.509466981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.25' to each side of the equation. 7.25 + -7.25 + x = -16.509466981 + -7.25 Combine like terms: 7.25 + -7.25 = 0.00 0.00 + x = -16.509466981 + -7.25 x = -16.509466981 + -7.25 Combine like terms: -16.509466981 + -7.25 = -23.759466981 x = -23.759466981 Simplifying x = -23.759466981Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.259466981, -23.759466981}Solution
x = {9.259466981, -23.759466981}
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