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Simplifying 7 + (a + -1)(a + 5) = 0 Reorder the terms: 7 + (-1 + a)(a + 5) = 0 Reorder the terms: 7 + (-1 + a)(5 + a) = 0 Multiply (-1 + a) * (5 + a) 7 + (-1(5 + a) + a(5 + a)) = 0 7 + ((5 * -1 + a * -1) + a(5 + a)) = 0 7 + ((-5 + -1a) + a(5 + a)) = 0 7 + (-5 + -1a + (5 * a + a * a)) = 0 7 + (-5 + -1a + (5a + a2)) = 0 Combine like terms: -1a + 5a = 4a 7 + (-5 + 4a + a2) = 0 Combine like terms: 7 + -5 = 2 2 + 4a + a2 = 0 Solving 2 + 4a + a2 = 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 = -0.585786438 a = -0.585786438 Simplifying a = -0.585786438Subproblem 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 = -3.414213562 a = -3.414213562 Simplifying a = -3.414213562Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.585786438, -3.414213562}
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