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Simplifying (5x + 62)(3x + 47) * 37 = 180 Reorder the terms: (62 + 5x)(3x + 47) * 37 = 180 Reorder the terms: (62 + 5x)(47 + 3x) * 37 = 180 Reorder the terms for easier multiplication: 37(62 + 5x)(47 + 3x) = 180 Multiply (62 + 5x) * (47 + 3x) 37(62(47 + 3x) + 5x * (47 + 3x)) = 180 37((47 * 62 + 3x * 62) + 5x * (47 + 3x)) = 180 37((2914 + 186x) + 5x * (47 + 3x)) = 180 37(2914 + 186x + (47 * 5x + 3x * 5x)) = 180 37(2914 + 186x + (235x + 15x2)) = 180 Combine like terms: 186x + 235x = 421x 37(2914 + 421x + 15x2) = 180 (2914 * 37 + 421x * 37 + 15x2 * 37) = 180 (107818 + 15577x + 555x2) = 180 Solving 107818 + 15577x + 555x2 = 180 Solving for variable 'x'. Reorder the terms: 107818 + -180 + 15577x + 555x2 = 180 + -180 Combine like terms: 107818 + -180 = 107638 107638 + 15577x + 555x2 = 180 + -180 Combine like terms: 180 + -180 = 0 107638 + 15577x + 555x2 = 0 Begin completing the square. Divide all terms by 555 the coefficient of the squared term: Divide each side by '555'. 193.9423423 + 28.06666667x + x2 = 0 Move the constant term to the right: Add '-193.9423423' to each side of the equation. 193.9423423 + 28.06666667x + -193.9423423 + x2 = 0 + -193.9423423 Reorder the terms: 193.9423423 + -193.9423423 + 28.06666667x + x2 = 0 + -193.9423423 Combine like terms: 193.9423423 + -193.9423423 = 0.0000000 0.0000000 + 28.06666667x + x2 = 0 + -193.9423423 28.06666667x + x2 = 0 + -193.9423423 Combine like terms: 0 + -193.9423423 = -193.9423423 28.06666667x + x2 = -193.9423423 The x term is 28.06666667x. Take half its coefficient (14.03333334). Square it (196.9344446) and add it to both sides. Add '196.9344446' to each side of the equation. 28.06666667x + 196.9344446 + x2 = -193.9423423 + 196.9344446 Reorder the terms: 196.9344446 + 28.06666667x + x2 = -193.9423423 + 196.9344446 Combine like terms: -193.9423423 + 196.9344446 = 2.9921023 196.9344446 + 28.06666667x + x2 = 2.9921023 Factor a perfect square on the left side: (x + 14.03333334)(x + 14.03333334) = 2.9921023 Calculate the square root of the right side: 1.729769436 Break this problem into two subproblems by setting (x + 14.03333334) equal to 1.729769436 and -1.729769436.Subproblem 1
x + 14.03333334 = 1.729769436 Simplifying x + 14.03333334 = 1.729769436 Reorder the terms: 14.03333334 + x = 1.729769436 Solving 14.03333334 + x = 1.729769436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-14.03333334' to each side of the equation. 14.03333334 + -14.03333334 + x = 1.729769436 + -14.03333334 Combine like terms: 14.03333334 + -14.03333334 = 0.00000000 0.00000000 + x = 1.729769436 + -14.03333334 x = 1.729769436 + -14.03333334 Combine like terms: 1.729769436 + -14.03333334 = -12.303563904 x = -12.303563904 Simplifying x = -12.303563904Subproblem 2
x + 14.03333334 = -1.729769436 Simplifying x + 14.03333334 = -1.729769436 Reorder the terms: 14.03333334 + x = -1.729769436 Solving 14.03333334 + x = -1.729769436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-14.03333334' to each side of the equation. 14.03333334 + -14.03333334 + x = -1.729769436 + -14.03333334 Combine like terms: 14.03333334 + -14.03333334 = 0.00000000 0.00000000 + x = -1.729769436 + -14.03333334 x = -1.729769436 + -14.03333334 Combine like terms: -1.729769436 + -14.03333334 = -15.763102776 x = -15.763102776 Simplifying x = -15.763102776Solution
The solution to the problem is based on the solutions from the subproblems. x = {-12.303563904, -15.763102776}
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