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Simplifying 5a(a + 3) + -1a = -1(4a + -6) + 1 Reorder the terms: 5a(3 + a) + -1a = -1(4a + -6) + 1 (3 * 5a + a * 5a) + -1a = -1(4a + -6) + 1 (15a + 5a2) + -1a = -1(4a + -6) + 1 Reorder the terms: 15a + -1a + 5a2 = -1(4a + -6) + 1 Combine like terms: 15a + -1a = 14a 14a + 5a2 = -1(4a + -6) + 1 Reorder the terms: 14a + 5a2 = -1(-6 + 4a) + 1 14a + 5a2 = (-6 * -1 + 4a * -1) + 1 14a + 5a2 = (6 + -4a) + 1 Reorder the terms: 14a + 5a2 = 6 + 1 + -4a Combine like terms: 6 + 1 = 7 14a + 5a2 = 7 + -4a Solving 14a + 5a2 = 7 + -4a Solving for variable 'a'. Reorder the terms: -7 + 14a + 4a + 5a2 = 7 + -4a + -7 + 4a Combine like terms: 14a + 4a = 18a -7 + 18a + 5a2 = 7 + -4a + -7 + 4a Reorder the terms: -7 + 18a + 5a2 = 7 + -7 + -4a + 4a Combine like terms: 7 + -7 = 0 -7 + 18a + 5a2 = 0 + -4a + 4a -7 + 18a + 5a2 = -4a + 4a Combine like terms: -4a + 4a = 0 -7 + 18a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.4 + 3.6a + a2 = 0 Move the constant term to the right: Add '1.4' to each side of the equation. -1.4 + 3.6a + 1.4 + a2 = 0 + 1.4 Reorder the terms: -1.4 + 1.4 + 3.6a + a2 = 0 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + 3.6a + a2 = 0 + 1.4 3.6a + a2 = 0 + 1.4 Combine like terms: 0 + 1.4 = 1.4 3.6a + a2 = 1.4 The a term is 3.6a. Take half its coefficient (1.8). Square it (3.24) and add it to both sides. Add '3.24' to each side of the equation. 3.6a + 3.24 + a2 = 1.4 + 3.24 Reorder the terms: 3.24 + 3.6a + a2 = 1.4 + 3.24 Combine like terms: 1.4 + 3.24 = 4.64 3.24 + 3.6a + a2 = 4.64 Factor a perfect square on the left side: (a + 1.8)(a + 1.8) = 4.64 Calculate the square root of the right side: 2.154065923 Break this problem into two subproblems by setting (a + 1.8) equal to 2.154065923 and -2.154065923.Subproblem 1
a + 1.8 = 2.154065923 Simplifying a + 1.8 = 2.154065923 Reorder the terms: 1.8 + a = 2.154065923 Solving 1.8 + a = 2.154065923 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.8' to each side of the equation. 1.8 + -1.8 + a = 2.154065923 + -1.8 Combine like terms: 1.8 + -1.8 = 0.0 0.0 + a = 2.154065923 + -1.8 a = 2.154065923 + -1.8 Combine like terms: 2.154065923 + -1.8 = 0.354065923 a = 0.354065923 Simplifying a = 0.354065923Subproblem 2
a + 1.8 = -2.154065923 Simplifying a + 1.8 = -2.154065923 Reorder the terms: 1.8 + a = -2.154065923 Solving 1.8 + a = -2.154065923 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.8' to each side of the equation. 1.8 + -1.8 + a = -2.154065923 + -1.8 Combine like terms: 1.8 + -1.8 = 0.0 0.0 + a = -2.154065923 + -1.8 a = -2.154065923 + -1.8 Combine like terms: -2.154065923 + -1.8 = -3.954065923 a = -3.954065923 Simplifying a = -3.954065923Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.354065923, -3.954065923}
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