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Simplifying 180 = (2x + 9)(3x + 1) Reorder the terms: 180 = (9 + 2x)(3x + 1) Reorder the terms: 180 = (9 + 2x)(1 + 3x) Multiply (9 + 2x) * (1 + 3x) 180 = (9(1 + 3x) + 2x * (1 + 3x)) 180 = ((1 * 9 + 3x * 9) + 2x * (1 + 3x)) 180 = ((9 + 27x) + 2x * (1 + 3x)) 180 = (9 + 27x + (1 * 2x + 3x * 2x)) 180 = (9 + 27x + (2x + 6x2)) Combine like terms: 27x + 2x = 29x 180 = (9 + 29x + 6x2) Solving 180 = 9 + 29x + 6x2 Solving for variable 'x'. Combine like terms: 180 + -9 = 171 171 + -29x + -6x2 = 9 + 29x + 6x2 + -9 + -29x + -6x2 Reorder the terms: 171 + -29x + -6x2 = 9 + -9 + 29x + -29x + 6x2 + -6x2 Combine like terms: 9 + -9 = 0 171 + -29x + -6x2 = 0 + 29x + -29x + 6x2 + -6x2 171 + -29x + -6x2 = 29x + -29x + 6x2 + -6x2 Combine like terms: 29x + -29x = 0 171 + -29x + -6x2 = 0 + 6x2 + -6x2 171 + -29x + -6x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 171 + -29x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -28.5 + 4.833333333x + x2 = 0 Move the constant term to the right: Add '28.5' to each side of the equation. -28.5 + 4.833333333x + 28.5 + x2 = 0 + 28.5 Reorder the terms: -28.5 + 28.5 + 4.833333333x + x2 = 0 + 28.5 Combine like terms: -28.5 + 28.5 = 0.0 0.0 + 4.833333333x + x2 = 0 + 28.5 4.833333333x + x2 = 0 + 28.5 Combine like terms: 0 + 28.5 = 28.5 4.833333333x + x2 = 28.5 The x term is 4.833333333x. Take half its coefficient (2.416666667). Square it (5.840277779) and add it to both sides. Add '5.840277779' to each side of the equation. 4.833333333x + 5.840277779 + x2 = 28.5 + 5.840277779 Reorder the terms: 5.840277779 + 4.833333333x + x2 = 28.5 + 5.840277779 Combine like terms: 28.5 + 5.840277779 = 34.340277779 5.840277779 + 4.833333333x + x2 = 34.340277779 Factor a perfect square on the left side: (x + 2.416666667)(x + 2.416666667) = 34.340277779 Calculate the square root of the right side: 5.860057831 Break this problem into two subproblems by setting (x + 2.416666667) equal to 5.860057831 and -5.860057831.Subproblem 1
x + 2.416666667 = 5.860057831 Simplifying x + 2.416666667 = 5.860057831 Reorder the terms: 2.416666667 + x = 5.860057831 Solving 2.416666667 + x = 5.860057831 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + x = 5.860057831 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + x = 5.860057831 + -2.416666667 x = 5.860057831 + -2.416666667 Combine like terms: 5.860057831 + -2.416666667 = 3.443391164 x = 3.443391164 Simplifying x = 3.443391164Subproblem 2
x + 2.416666667 = -5.860057831 Simplifying x + 2.416666667 = -5.860057831 Reorder the terms: 2.416666667 + x = -5.860057831 Solving 2.416666667 + x = -5.860057831 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + x = -5.860057831 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + x = -5.860057831 + -2.416666667 x = -5.860057831 + -2.416666667 Combine like terms: -5.860057831 + -2.416666667 = -8.276724498 x = -8.276724498 Simplifying x = -8.276724498Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.443391164, -8.276724498}
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