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Simplifying (a + -3)(a + -1) = 2 Reorder the terms: (-3 + a)(a + -1) = 2 Reorder the terms: (-3 + a)(-1 + a) = 2 Multiply (-3 + a) * (-1 + a) (-3(-1 + a) + a(-1 + a)) = 2 ((-1 * -3 + a * -3) + a(-1 + a)) = 2 ((3 + -3a) + a(-1 + a)) = 2 (3 + -3a + (-1 * a + a * a)) = 2 (3 + -3a + (-1a + a2)) = 2 Combine like terms: -3a + -1a = -4a (3 + -4a + a2) = 2 Solving 3 + -4a + a2 = 2 Solving for variable 'a'. Reorder the terms: 3 + -2 + -4a + a2 = 2 + -2 Combine like terms: 3 + -2 = 1 1 + -4a + a2 = 2 + -2 Combine like terms: 2 + -2 = 0 1 + -4a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -4a + -1 + a2 = 0 + -1 Reorder the terms: 1 + -1 + -4a + a2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4a + a2 = 0 + -1 -4a + a2 = 0 + -1 Combine like terms: 0 + -1 = -1 -4a + a2 = -1 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 = -1 + 4 Reorder the terms: 4 + -4a + a2 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + -4a + a2 = 3 Factor a perfect square on the left side: (a + -2)(a + -2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (a + -2) equal to 1.732050808 and -1.732050808.Subproblem 1
a + -2 = 1.732050808 Simplifying a + -2 = 1.732050808 Reorder the terms: -2 + a = 1.732050808 Solving -2 + a = 1.732050808 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 = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + a = 1.732050808 + 2 a = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 a = 3.732050808 Simplifying a = 3.732050808Subproblem 2
a + -2 = -1.732050808 Simplifying a + -2 = -1.732050808 Reorder the terms: -2 + a = -1.732050808 Solving -2 + a = -1.732050808 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 = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + a = -1.732050808 + 2 a = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 a = 0.267949192 Simplifying a = 0.267949192Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.732050808, 0.267949192}
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