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Simplifying 2x(5 + 3x) + x = 32 (5 * 2x + 3x * 2x) + x = 32 (10x + 6x2) + x = 32 Reorder the terms: 10x + x + 6x2 = 32 Combine like terms: 10x + x = 11x 11x + 6x2 = 32 Solving 11x + 6x2 = 32 Solving for variable 'x'. Reorder the terms: -32 + 11x + 6x2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 11x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.333333333 + 1.833333333x + x2 = 0 Move the constant term to the right: Add '5.333333333' to each side of the equation. -5.333333333 + 1.833333333x + 5.333333333 + x2 = 0 + 5.333333333 Reorder the terms: -5.333333333 + 5.333333333 + 1.833333333x + x2 = 0 + 5.333333333 Combine like terms: -5.333333333 + 5.333333333 = 0.000000000 0.000000000 + 1.833333333x + x2 = 0 + 5.333333333 1.833333333x + x2 = 0 + 5.333333333 Combine like terms: 0 + 5.333333333 = 5.333333333 1.833333333x + x2 = 5.333333333 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 = 5.333333333 + 0.8402777775 Reorder the terms: 0.8402777775 + 1.833333333x + x2 = 5.333333333 + 0.8402777775 Combine like terms: 5.333333333 + 0.8402777775 = 6.1736111105 0.8402777775 + 1.833333333x + x2 = 6.1736111105 Factor a perfect square on the left side: (x + 0.9166666665)(x + 0.9166666665) = 6.1736111105 Calculate the square root of the right side: 2.484675253 Break this problem into two subproblems by setting (x + 0.9166666665) equal to 2.484675253 and -2.484675253.Subproblem 1
x + 0.9166666665 = 2.484675253 Simplifying x + 0.9166666665 = 2.484675253 Reorder the terms: 0.9166666665 + x = 2.484675253 Solving 0.9166666665 + x = 2.484675253 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 = 2.484675253 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = 2.484675253 + -0.9166666665 x = 2.484675253 + -0.9166666665 Combine like terms: 2.484675253 + -0.9166666665 = 1.5680085865 x = 1.5680085865 Simplifying x = 1.5680085865Subproblem 2
x + 0.9166666665 = -2.484675253 Simplifying x + 0.9166666665 = -2.484675253 Reorder the terms: 0.9166666665 + x = -2.484675253 Solving 0.9166666665 + x = -2.484675253 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 = -2.484675253 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = -2.484675253 + -0.9166666665 x = -2.484675253 + -0.9166666665 Combine like terms: -2.484675253 + -0.9166666665 = -3.4013419195 x = -3.4013419195 Simplifying x = -3.4013419195Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5680085865, -3.4013419195}
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