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Simplifying (-1a + -3) + -1(3a + -1a2 + -5) = 0 Reorder the terms: (-3 + -1a) + -1(3a + -1a2 + -5) = 0 Remove parenthesis around (-3 + -1a) -3 + -1a + -1(3a + -1a2 + -5) = 0 Reorder the terms: -3 + -1a + -1(-5 + 3a + -1a2) = 0 -3 + -1a + (-5 * -1 + 3a * -1 + -1a2 * -1) = 0 -3 + -1a + (5 + -3a + 1a2) = 0 Reorder the terms: -3 + 5 + -1a + -3a + 1a2 = 0 Combine like terms: -3 + 5 = 2 2 + -1a + -3a + 1a2 = 0 Combine like terms: -1a + -3a = -4a 2 + -4a + 1a2 = 0 Solving 2 + -4a + 1a2 = 0 Solving for variable 'a'. Begin completing the square. Move the constant term to the right: Add '-2' to each side of the equation. 2 + -4a + -2 + a2 = 0 + -2 Reorder the terms: 2 + -2 + -4a + a2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -4a + a2 = 0 + -2 -4a + a2 = 0 + -2 Combine like terms: 0 + -2 = -2 -4a + a2 = -2 The a term is -4a. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4a + 4 + a2 = -2 + 4 Reorder the terms: 4 + -4a + a2 = -2 + 4 Combine like terms: -2 + 4 = 2 4 + -4a + a2 = 2 Factor a perfect square on the left side: (a + -2)(a + -2) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (a + -2) equal to 1.414213562 and -1.414213562.Subproblem 1
a + -2 = 1.414213562 Simplifying a + -2 = 1.414213562 Reorder the terms: -2 + a = 1.414213562 Solving -2 + a = 1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + a = 1.414213562 + 2 a = 1.414213562 + 2 Combine like terms: 1.414213562 + 2 = 3.414213562 a = 3.414213562 Simplifying a = 3.414213562Subproblem 2
a + -2 = -1.414213562 Simplifying a + -2 = -1.414213562 Reorder the terms: -2 + a = -1.414213562 Solving -2 + a = -1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + a = -1.414213562 + 2 a = -1.414213562 + 2 Combine like terms: -1.414213562 + 2 = 0.585786438 a = 0.585786438 Simplifying a = 0.585786438Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.414213562, 0.585786438}
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