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Simplifying (2x + 1)(3x + 4) = 180 Reorder the terms: (1 + 2x)(3x + 4) = 180 Reorder the terms: (1 + 2x)(4 + 3x) = 180 Multiply (1 + 2x) * (4 + 3x) (1(4 + 3x) + 2x * (4 + 3x)) = 180 ((4 * 1 + 3x * 1) + 2x * (4 + 3x)) = 180 ((4 + 3x) + 2x * (4 + 3x)) = 180 (4 + 3x + (4 * 2x + 3x * 2x)) = 180 (4 + 3x + (8x + 6x2)) = 180 Combine like terms: 3x + 8x = 11x (4 + 11x + 6x2) = 180 Solving 4 + 11x + 6x2 = 180 Solving for variable 'x'. Reorder the terms: 4 + -180 + 11x + 6x2 = 180 + -180 Combine like terms: 4 + -180 = -176 -176 + 11x + 6x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -176 + 11x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -29.33333333 + 1.833333333x + x2 = 0 Move the constant term to the right: Add '29.33333333' to each side of the equation. -29.33333333 + 1.833333333x + 29.33333333 + x2 = 0 + 29.33333333 Reorder the terms: -29.33333333 + 29.33333333 + 1.833333333x + x2 = 0 + 29.33333333 Combine like terms: -29.33333333 + 29.33333333 = 0.00000000 0.00000000 + 1.833333333x + x2 = 0 + 29.33333333 1.833333333x + x2 = 0 + 29.33333333 Combine like terms: 0 + 29.33333333 = 29.33333333 1.833333333x + x2 = 29.33333333 The x term is 1.833333333x. Take half its coefficient (0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. 1.833333333x + 0.8402777775 + x2 = 29.33333333 + 0.8402777775 Reorder the terms: 0.8402777775 + 1.833333333x + x2 = 29.33333333 + 0.8402777775 Combine like terms: 29.33333333 + 0.8402777775 = 30.1736111075 0.8402777775 + 1.833333333x + x2 = 30.1736111075 Factor a perfect square on the left side: (x + 0.9166666665)(x + 0.9166666665) = 30.1736111075 Calculate the square root of the right side: 5.493051166 Break this problem into two subproblems by setting (x + 0.9166666665) equal to 5.493051166 and -5.493051166.Subproblem 1
x + 0.9166666665 = 5.493051166 Simplifying x + 0.9166666665 = 5.493051166 Reorder the terms: 0.9166666665 + x = 5.493051166 Solving 0.9166666665 + x = 5.493051166 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = 5.493051166 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = 5.493051166 + -0.9166666665 x = 5.493051166 + -0.9166666665 Combine like terms: 5.493051166 + -0.9166666665 = 4.5763844995 x = 4.5763844995 Simplifying x = 4.5763844995Subproblem 2
x + 0.9166666665 = -5.493051166 Simplifying x + 0.9166666665 = -5.493051166 Reorder the terms: 0.9166666665 + x = -5.493051166 Solving 0.9166666665 + x = -5.493051166 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = -5.493051166 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = -5.493051166 + -0.9166666665 x = -5.493051166 + -0.9166666665 Combine like terms: -5.493051166 + -0.9166666665 = -6.4097178325 x = -6.4097178325 Simplifying x = -6.4097178325Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.5763844995, -6.4097178325}
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