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Simplifying (3x + 20)(7x + 40) = 180 Reorder the terms: (20 + 3x)(7x + 40) = 180 Reorder the terms: (20 + 3x)(40 + 7x) = 180 Multiply (20 + 3x) * (40 + 7x) (20(40 + 7x) + 3x * (40 + 7x)) = 180 ((40 * 20 + 7x * 20) + 3x * (40 + 7x)) = 180 ((800 + 140x) + 3x * (40 + 7x)) = 180 (800 + 140x + (40 * 3x + 7x * 3x)) = 180 (800 + 140x + (120x + 21x2)) = 180 Combine like terms: 140x + 120x = 260x (800 + 260x + 21x2) = 180 Solving 800 + 260x + 21x2 = 180 Solving for variable 'x'. Reorder the terms: 800 + -180 + 260x + 21x2 = 180 + -180 Combine like terms: 800 + -180 = 620 620 + 260x + 21x2 = 180 + -180 Combine like terms: 180 + -180 = 0 620 + 260x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. 29.52380952 + 12.38095238x + x2 = 0 Move the constant term to the right: Add '-29.52380952' to each side of the equation. 29.52380952 + 12.38095238x + -29.52380952 + x2 = 0 + -29.52380952 Reorder the terms: 29.52380952 + -29.52380952 + 12.38095238x + x2 = 0 + -29.52380952 Combine like terms: 29.52380952 + -29.52380952 = 0.00000000 0.00000000 + 12.38095238x + x2 = 0 + -29.52380952 12.38095238x + x2 = 0 + -29.52380952 Combine like terms: 0 + -29.52380952 = -29.52380952 12.38095238x + x2 = -29.52380952 The x term is 12.38095238x. Take half its coefficient (6.19047619). Square it (38.32199546) and add it to both sides. Add '38.32199546' to each side of the equation. 12.38095238x + 38.32199546 + x2 = -29.52380952 + 38.32199546 Reorder the terms: 38.32199546 + 12.38095238x + x2 = -29.52380952 + 38.32199546 Combine like terms: -29.52380952 + 38.32199546 = 8.79818594 38.32199546 + 12.38095238x + x2 = 8.79818594 Factor a perfect square on the left side: (x + 6.19047619)(x + 6.19047619) = 8.79818594 Calculate the square root of the right side: 2.966173619 Break this problem into two subproblems by setting (x + 6.19047619) equal to 2.966173619 and -2.966173619.Subproblem 1
x + 6.19047619 = 2.966173619 Simplifying x + 6.19047619 = 2.966173619 Reorder the terms: 6.19047619 + x = 2.966173619 Solving 6.19047619 + x = 2.966173619 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.19047619' to each side of the equation. 6.19047619 + -6.19047619 + x = 2.966173619 + -6.19047619 Combine like terms: 6.19047619 + -6.19047619 = 0.00000000 0.00000000 + x = 2.966173619 + -6.19047619 x = 2.966173619 + -6.19047619 Combine like terms: 2.966173619 + -6.19047619 = -3.224302571 x = -3.224302571 Simplifying x = -3.224302571Subproblem 2
x + 6.19047619 = -2.966173619 Simplifying x + 6.19047619 = -2.966173619 Reorder the terms: 6.19047619 + x = -2.966173619 Solving 6.19047619 + x = -2.966173619 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.19047619' to each side of the equation. 6.19047619 + -6.19047619 + x = -2.966173619 + -6.19047619 Combine like terms: 6.19047619 + -6.19047619 = 0.00000000 0.00000000 + x = -2.966173619 + -6.19047619 x = -2.966173619 + -6.19047619 Combine like terms: -2.966173619 + -6.19047619 = -9.156649809 x = -9.156649809 Simplifying x = -9.156649809Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.224302571, -9.156649809}
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