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Simplifying 5a(8 + 2a) + 5 = (-1a + 1) + -5 + 2 + -1 (8 * 5a + 2a * 5a) + 5 = (-1a + 1) + -5 + 2 + -1 (40a + 10a2) + 5 = (-1a + 1) + -5 + 2 + -1 Reorder the terms: 5 + 40a + 10a2 = (-1a + 1) + -5 + 2 + -1 Reorder the terms: 5 + 40a + 10a2 = (1 + -1a) + -5 + 2 + -1 Remove parenthesis around (1 + -1a) 5 + 40a + 10a2 = 1 + -1a + -5 + 2 + -1 Reorder the terms: 5 + 40a + 10a2 = 1 + -5 + 2 + -1 + -1a Combine like terms: 1 + -5 = -4 5 + 40a + 10a2 = -4 + 2 + -1 + -1a Combine like terms: -4 + 2 = -2 5 + 40a + 10a2 = -2 + -1 + -1a Combine like terms: -2 + -1 = -3 5 + 40a + 10a2 = -3 + -1a Solving 5 + 40a + 10a2 = -3 + -1a Solving for variable 'a'. Reorder the terms: 5 + 3 + 40a + a + 10a2 = -3 + -1a + 3 + a Combine like terms: 5 + 3 = 8 8 + 40a + a + 10a2 = -3 + -1a + 3 + a Combine like terms: 40a + a = 41a 8 + 41a + 10a2 = -3 + -1a + 3 + a Reorder the terms: 8 + 41a + 10a2 = -3 + 3 + -1a + a Combine like terms: -3 + 3 = 0 8 + 41a + 10a2 = 0 + -1a + a 8 + 41a + 10a2 = -1a + a Combine like terms: -1a + a = 0 8 + 41a + 10a2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.8 + 4.1a + a2 = 0.0 Move the constant term to the right: Add '-0.8' to each side of the equation. 0.8 + 4.1a + -0.8 + a2 = 0.0 + -0.8 Reorder the terms: 0.8 + -0.8 + 4.1a + a2 = 0.0 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + 4.1a + a2 = 0.0 + -0.8 4.1a + a2 = 0.0 + -0.8 Combine like terms: 0.0 + -0.8 = -0.8 4.1a + a2 = -0.8 The a term is 4.1a. Take half its coefficient (2.05). Square it (4.2025) and add it to both sides. Add '4.2025' to each side of the equation. 4.1a + 4.2025 + a2 = -0.8 + 4.2025 Reorder the terms: 4.2025 + 4.1a + a2 = -0.8 + 4.2025 Combine like terms: -0.8 + 4.2025 = 3.4025 4.2025 + 4.1a + a2 = 3.4025 Factor a perfect square on the left side: (a + 2.05)(a + 2.05) = 3.4025 Calculate the square root of the right side: 1.844586675 Break this problem into two subproblems by setting (a + 2.05) equal to 1.844586675 and -1.844586675.Subproblem 1
a + 2.05 = 1.844586675 Simplifying a + 2.05 = 1.844586675 Reorder the terms: 2.05 + a = 1.844586675 Solving 2.05 + a = 1.844586675 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + a = 1.844586675 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + a = 1.844586675 + -2.05 a = 1.844586675 + -2.05 Combine like terms: 1.844586675 + -2.05 = -0.205413325 a = -0.205413325 Simplifying a = -0.205413325Subproblem 2
a + 2.05 = -1.844586675 Simplifying a + 2.05 = -1.844586675 Reorder the terms: 2.05 + a = -1.844586675 Solving 2.05 + a = -1.844586675 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + a = -1.844586675 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + a = -1.844586675 + -2.05 a = -1.844586675 + -2.05 Combine like terms: -1.844586675 + -2.05 = -3.894586675 a = -3.894586675 Simplifying a = -3.894586675Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.205413325, -3.894586675}
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