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Simplifying (x + 2)(6x + 10) + x = 180 Reorder the terms: (2 + x)(6x + 10) + x = 180 Reorder the terms: (2 + x)(10 + 6x) + x = 180 Multiply (2 + x) * (10 + 6x) (2(10 + 6x) + x(10 + 6x)) + x = 180 ((10 * 2 + 6x * 2) + x(10 + 6x)) + x = 180 ((20 + 12x) + x(10 + 6x)) + x = 180 (20 + 12x + (10 * x + 6x * x)) + x = 180 (20 + 12x + (10x + 6x2)) + x = 180 Combine like terms: 12x + 10x = 22x (20 + 22x + 6x2) + x = 180 Reorder the terms: 20 + 22x + x + 6x2 = 180 Combine like terms: 22x + x = 23x 20 + 23x + 6x2 = 180 Solving 20 + 23x + 6x2 = 180 Solving for variable 'x'. Reorder the terms: 20 + -180 + 23x + 6x2 = 180 + -180 Combine like terms: 20 + -180 = -160 -160 + 23x + 6x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -160 + 23x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -26.66666667 + 3.833333333x + x2 = 0 Move the constant term to the right: Add '26.66666667' to each side of the equation. -26.66666667 + 3.833333333x + 26.66666667 + x2 = 0 + 26.66666667 Reorder the terms: -26.66666667 + 26.66666667 + 3.833333333x + x2 = 0 + 26.66666667 Combine like terms: -26.66666667 + 26.66666667 = 0.00000000 0.00000000 + 3.833333333x + x2 = 0 + 26.66666667 3.833333333x + x2 = 0 + 26.66666667 Combine like terms: 0 + 26.66666667 = 26.66666667 3.833333333x + x2 = 26.66666667 The x term is 3.833333333x. Take half its coefficient (1.916666667). Square it (3.673611112) and add it to both sides. Add '3.673611112' to each side of the equation. 3.833333333x + 3.673611112 + x2 = 26.66666667 + 3.673611112 Reorder the terms: 3.673611112 + 3.833333333x + x2 = 26.66666667 + 3.673611112 Combine like terms: 26.66666667 + 3.673611112 = 30.340277782 3.673611112 + 3.833333333x + x2 = 30.340277782 Factor a perfect square on the left side: (x + 1.916666667)(x + 1.916666667) = 30.340277782 Calculate the square root of the right side: 5.508200957 Break this problem into two subproblems by setting (x + 1.916666667) equal to 5.508200957 and -5.508200957.Subproblem 1
x + 1.916666667 = 5.508200957 Simplifying x + 1.916666667 = 5.508200957 Reorder the terms: 1.916666667 + x = 5.508200957 Solving 1.916666667 + x = 5.508200957 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = 5.508200957 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = 5.508200957 + -1.916666667 x = 5.508200957 + -1.916666667 Combine like terms: 5.508200957 + -1.916666667 = 3.59153429 x = 3.59153429 Simplifying x = 3.59153429Subproblem 2
x + 1.916666667 = -5.508200957 Simplifying x + 1.916666667 = -5.508200957 Reorder the terms: 1.916666667 + x = -5.508200957 Solving 1.916666667 + x = -5.508200957 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = -5.508200957 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = -5.508200957 + -1.916666667 x = -5.508200957 + -1.916666667 Combine like terms: -5.508200957 + -1.916666667 = -7.424867624 x = -7.424867624 Simplifying x = -7.424867624Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.59153429, -7.424867624}
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