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Simplifying 3a + 1 = 3.6a(a + -1) Reorder the terms: 1 + 3a = 3.6a(a + -1) Reorder the terms: 1 + 3a = 3.6a(-1 + a) 1 + 3a = (-1 * 3.6a + a * 3.6a) 1 + 3a = (-3.6a + 3.6a2) Solving 1 + 3a = -3.6a + 3.6a2 Solving for variable 'a'. Combine like terms: 3a + 3.6a = 6.6a 1 + 6.6a + -3.6a2 = -3.6a + 3.6a2 + 3.6a + -3.6a2 Reorder the terms: 1 + 6.6a + -3.6a2 = -3.6a + 3.6a + 3.6a2 + -3.6a2 Combine like terms: -3.6a + 3.6a = 0.0 1 + 6.6a + -3.6a2 = 0.0 + 3.6a2 + -3.6a2 1 + 6.6a + -3.6a2 = 3.6a2 + -3.6a2 Combine like terms: 3.6a2 + -3.6a2 = 0.0 1 + 6.6a + -3.6a2 = 0.0 Begin completing the square. Divide all terms by -3.6 the coefficient of the squared term: Divide each side by '-3.6'. -0.2777777778 + -1.833333333a + a2 = 0 Move the constant term to the right: Add '0.2777777778' to each side of the equation. -0.2777777778 + -1.833333333a + 0.2777777778 + a2 = 0 + 0.2777777778 Reorder the terms: -0.2777777778 + 0.2777777778 + -1.833333333a + a2 = 0 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + -1.833333333a + a2 = 0 + 0.2777777778 -1.833333333a + a2 = 0 + 0.2777777778 Combine like terms: 0 + 0.2777777778 = 0.2777777778 -1.833333333a + a2 = 0.2777777778 The a term is -1.833333333a. 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. -1.833333333a + 0.8402777775 + a2 = 0.2777777778 + 0.8402777775 Reorder the terms: 0.8402777775 + -1.833333333a + a2 = 0.2777777778 + 0.8402777775 Combine like terms: 0.2777777778 + 0.8402777775 = 1.1180555553 0.8402777775 + -1.833333333a + a2 = 1.1180555553 Factor a perfect square on the left side: (a + -0.9166666665)(a + -0.9166666665) = 1.1180555553 Calculate the square root of the right side: 1.057381462 Break this problem into two subproblems by setting (a + -0.9166666665) equal to 1.057381462 and -1.057381462.Subproblem 1
a + -0.9166666665 = 1.057381462 Simplifying a + -0.9166666665 = 1.057381462 Reorder the terms: -0.9166666665 + a = 1.057381462 Solving -0.9166666665 + a = 1.057381462 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + a = 1.057381462 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + a = 1.057381462 + 0.9166666665 a = 1.057381462 + 0.9166666665 Combine like terms: 1.057381462 + 0.9166666665 = 1.9740481285 a = 1.9740481285 Simplifying a = 1.9740481285Subproblem 2
a + -0.9166666665 = -1.057381462 Simplifying a + -0.9166666665 = -1.057381462 Reorder the terms: -0.9166666665 + a = -1.057381462 Solving -0.9166666665 + a = -1.057381462 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + a = -1.057381462 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + a = -1.057381462 + 0.9166666665 a = -1.057381462 + 0.9166666665 Combine like terms: -1.057381462 + 0.9166666665 = -0.1407147955 a = -0.1407147955 Simplifying a = -0.1407147955Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.9740481285, -0.1407147955}
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