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Simplifying (7x + -115)(3x + -35) = 180 Reorder the terms: (-115 + 7x)(3x + -35) = 180 Reorder the terms: (-115 + 7x)(-35 + 3x) = 180 Multiply (-115 + 7x) * (-35 + 3x) (-115(-35 + 3x) + 7x * (-35 + 3x)) = 180 ((-35 * -115 + 3x * -115) + 7x * (-35 + 3x)) = 180 ((4025 + -345x) + 7x * (-35 + 3x)) = 180 (4025 + -345x + (-35 * 7x + 3x * 7x)) = 180 (4025 + -345x + (-245x + 21x2)) = 180 Combine like terms: -345x + -245x = -590x (4025 + -590x + 21x2) = 180 Solving 4025 + -590x + 21x2 = 180 Solving for variable 'x'. Reorder the terms: 4025 + -180 + -590x + 21x2 = 180 + -180 Combine like terms: 4025 + -180 = 3845 3845 + -590x + 21x2 = 180 + -180 Combine like terms: 180 + -180 = 0 3845 + -590x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. 183.0952381 + -28.0952381x + x2 = 0 Move the constant term to the right: Add '-183.0952381' to each side of the equation. 183.0952381 + -28.0952381x + -183.0952381 + x2 = 0 + -183.0952381 Reorder the terms: 183.0952381 + -183.0952381 + -28.0952381x + x2 = 0 + -183.0952381 Combine like terms: 183.0952381 + -183.0952381 = 0.0000000 0.0000000 + -28.0952381x + x2 = 0 + -183.0952381 -28.0952381x + x2 = 0 + -183.0952381 Combine like terms: 0 + -183.0952381 = -183.0952381 -28.0952381x + x2 = -183.0952381 The x term is -28.0952381x. Take half its coefficient (-14.04761905). Square it (197.3356010) and add it to both sides. Add '197.3356010' to each side of the equation. -28.0952381x + 197.3356010 + x2 = -183.0952381 + 197.3356010 Reorder the terms: 197.3356010 + -28.0952381x + x2 = -183.0952381 + 197.3356010 Combine like terms: -183.0952381 + 197.3356010 = 14.2403629 197.3356010 + -28.0952381x + x2 = 14.2403629 Factor a perfect square on the left side: (x + -14.04761905)(x + -14.04761905) = 14.2403629 Calculate the square root of the right side: 3.773640537 Break this problem into two subproblems by setting (x + -14.04761905) equal to 3.773640537 and -3.773640537.Subproblem 1
x + -14.04761905 = 3.773640537 Simplifying x + -14.04761905 = 3.773640537 Reorder the terms: -14.04761905 + x = 3.773640537 Solving -14.04761905 + x = 3.773640537 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.04761905' to each side of the equation. -14.04761905 + 14.04761905 + x = 3.773640537 + 14.04761905 Combine like terms: -14.04761905 + 14.04761905 = 0.00000000 0.00000000 + x = 3.773640537 + 14.04761905 x = 3.773640537 + 14.04761905 Combine like terms: 3.773640537 + 14.04761905 = 17.821259587 x = 17.821259587 Simplifying x = 17.821259587Subproblem 2
x + -14.04761905 = -3.773640537 Simplifying x + -14.04761905 = -3.773640537 Reorder the terms: -14.04761905 + x = -3.773640537 Solving -14.04761905 + x = -3.773640537 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.04761905' to each side of the equation. -14.04761905 + 14.04761905 + x = -3.773640537 + 14.04761905 Combine like terms: -14.04761905 + 14.04761905 = 0.00000000 0.00000000 + x = -3.773640537 + 14.04761905 x = -3.773640537 + 14.04761905 Combine like terms: -3.773640537 + 14.04761905 = 10.273978513 x = 10.273978513 Simplifying x = 10.273978513Solution
The solution to the problem is based on the solutions from the subproblems. x = {17.821259587, 10.273978513}
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