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Simplifying (2x + 17)(3x + 7) = 3575 Reorder the terms: (17 + 2x)(3x + 7) = 3575 Reorder the terms: (17 + 2x)(7 + 3x) = 3575 Multiply (17 + 2x) * (7 + 3x) (17(7 + 3x) + 2x * (7 + 3x)) = 3575 ((7 * 17 + 3x * 17) + 2x * (7 + 3x)) = 3575 ((119 + 51x) + 2x * (7 + 3x)) = 3575 (119 + 51x + (7 * 2x + 3x * 2x)) = 3575 (119 + 51x + (14x + 6x2)) = 3575 Combine like terms: 51x + 14x = 65x (119 + 65x + 6x2) = 3575 Solving 119 + 65x + 6x2 = 3575 Solving for variable 'x'. Reorder the terms: 119 + -3575 + 65x + 6x2 = 3575 + -3575 Combine like terms: 119 + -3575 = -3456 -3456 + 65x + 6x2 = 3575 + -3575 Combine like terms: 3575 + -3575 = 0 -3456 + 65x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -576 + 10.83333333x + x2 = 0 Move the constant term to the right: Add '576' to each side of the equation. -576 + 10.83333333x + 576 + x2 = 0 + 576 Reorder the terms: -576 + 576 + 10.83333333x + x2 = 0 + 576 Combine like terms: -576 + 576 = 0 0 + 10.83333333x + x2 = 0 + 576 10.83333333x + x2 = 0 + 576 Combine like terms: 0 + 576 = 576 10.83333333x + x2 = 576 The x term is 10.83333333x. Take half its coefficient (5.416666665). Square it (29.34027776) and add it to both sides. Add '29.34027776' to each side of the equation. 10.83333333x + 29.34027776 + x2 = 576 + 29.34027776 Reorder the terms: 29.34027776 + 10.83333333x + x2 = 576 + 29.34027776 Combine like terms: 576 + 29.34027776 = 605.34027776 29.34027776 + 10.83333333x + x2 = 605.34027776 Factor a perfect square on the left side: (x + 5.416666665)(x + 5.416666665) = 605.34027776 Calculate the square root of the right side: 24.603663909 Break this problem into two subproblems by setting (x + 5.416666665) equal to 24.603663909 and -24.603663909.Subproblem 1
x + 5.416666665 = 24.603663909 Simplifying x + 5.416666665 = 24.603663909 Reorder the terms: 5.416666665 + x = 24.603663909 Solving 5.416666665 + x = 24.603663909 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.416666665' to each side of the equation. 5.416666665 + -5.416666665 + x = 24.603663909 + -5.416666665 Combine like terms: 5.416666665 + -5.416666665 = 0.000000000 0.000000000 + x = 24.603663909 + -5.416666665 x = 24.603663909 + -5.416666665 Combine like terms: 24.603663909 + -5.416666665 = 19.186997244 x = 19.186997244 Simplifying x = 19.186997244Subproblem 2
x + 5.416666665 = -24.603663909 Simplifying x + 5.416666665 = -24.603663909 Reorder the terms: 5.416666665 + x = -24.603663909 Solving 5.416666665 + x = -24.603663909 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.416666665' to each side of the equation. 5.416666665 + -5.416666665 + x = -24.603663909 + -5.416666665 Combine like terms: 5.416666665 + -5.416666665 = 0.000000000 0.000000000 + x = -24.603663909 + -5.416666665 x = -24.603663909 + -5.416666665 Combine like terms: -24.603663909 + -5.416666665 = -30.020330574 x = -30.020330574 Simplifying x = -30.020330574Solution
The solution to the problem is based on the solutions from the subproblems. x = {19.186997244, -30.020330574}
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