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Simplifying 5a(8 + 2a) + 5 = (-1a + 1) + -5 + 12 (8 * 5a + 2a * 5a) + 5 = (-1a + 1) + -5 + 12 (40a + 10a2) + 5 = (-1a + 1) + -5 + 12 Reorder the terms: 5 + 40a + 10a2 = (-1a + 1) + -5 + 12 Reorder the terms: 5 + 40a + 10a2 = (1 + -1a) + -5 + 12 Remove parenthesis around (1 + -1a) 5 + 40a + 10a2 = 1 + -1a + -5 + 12 Reorder the terms: 5 + 40a + 10a2 = 1 + -5 + 12 + -1a Combine like terms: 1 + -5 = -4 5 + 40a + 10a2 = -4 + 12 + -1a Combine like terms: -4 + 12 = 8 5 + 40a + 10a2 = 8 + -1a Solving 5 + 40a + 10a2 = 8 + -1a Solving for variable 'a'. Reorder the terms: 5 + -8 + 40a + a + 10a2 = 8 + -1a + -8 + a Combine like terms: 5 + -8 = -3 -3 + 40a + a + 10a2 = 8 + -1a + -8 + a Combine like terms: 40a + a = 41a -3 + 41a + 10a2 = 8 + -1a + -8 + a Reorder the terms: -3 + 41a + 10a2 = 8 + -8 + -1a + a Combine like terms: 8 + -8 = 0 -3 + 41a + 10a2 = 0 + -1a + a -3 + 41a + 10a2 = -1a + a Combine like terms: -1a + a = 0 -3 + 41a + 10a2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.3 + 4.1a + a2 = 0.0 Move the constant term to the right: Add '0.3' to each side of the equation. -0.3 + 4.1a + 0.3 + a2 = 0.0 + 0.3 Reorder the terms: -0.3 + 0.3 + 4.1a + a2 = 0.0 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + 4.1a + a2 = 0.0 + 0.3 4.1a + a2 = 0.0 + 0.3 Combine like terms: 0.0 + 0.3 = 0.3 4.1a + a2 = 0.3 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 = 0.3 + 4.2025 Reorder the terms: 4.2025 + 4.1a + a2 = 0.3 + 4.2025 Combine like terms: 0.3 + 4.2025 = 4.5025 4.2025 + 4.1a + a2 = 4.5025 Factor a perfect square on the left side: (a + 2.05)(a + 2.05) = 4.5025 Calculate the square root of the right side: 2.121909517 Break this problem into two subproblems by setting (a + 2.05) equal to 2.121909517 and -2.121909517.Subproblem 1
a + 2.05 = 2.121909517 Simplifying a + 2.05 = 2.121909517 Reorder the terms: 2.05 + a = 2.121909517 Solving 2.05 + a = 2.121909517 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 = 2.121909517 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + a = 2.121909517 + -2.05 a = 2.121909517 + -2.05 Combine like terms: 2.121909517 + -2.05 = 0.071909517 a = 0.071909517 Simplifying a = 0.071909517Subproblem 2
a + 2.05 = -2.121909517 Simplifying a + 2.05 = -2.121909517 Reorder the terms: 2.05 + a = -2.121909517 Solving 2.05 + a = -2.121909517 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 = -2.121909517 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + a = -2.121909517 + -2.05 a = -2.121909517 + -2.05 Combine like terms: -2.121909517 + -2.05 = -4.171909517 a = -4.171909517 Simplifying a = -4.171909517Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.071909517, -4.171909517}
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