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Simplifying a2 + 12a + 23 = 7 Reorder the terms: 23 + 12a + a2 = 7 Solving 23 + 12a + a2 = 7 Solving for variable 'a'. Reorder the terms: 23 + -7 + 12a + a2 = 7 + -7 Combine like terms: 23 + -7 = 16 16 + 12a + a2 = 7 + -7 Combine like terms: 7 + -7 = 0 16 + 12a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-16' to each side of the equation. 16 + 12a + -16 + a2 = 0 + -16 Reorder the terms: 16 + -16 + 12a + a2 = 0 + -16 Combine like terms: 16 + -16 = 0 0 + 12a + a2 = 0 + -16 12a + a2 = 0 + -16 Combine like terms: 0 + -16 = -16 12a + a2 = -16 The a term is 12a. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12a + 36 + a2 = -16 + 36 Reorder the terms: 36 + 12a + a2 = -16 + 36 Combine like terms: -16 + 36 = 20 36 + 12a + a2 = 20 Factor a perfect square on the left side: (a + 6)(a + 6) = 20 Calculate the square root of the right side: 4.472135955 Break this problem into two subproblems by setting (a + 6) equal to 4.472135955 and -4.472135955.Subproblem 1
a + 6 = 4.472135955 Simplifying a + 6 = 4.472135955 Reorder the terms: 6 + a = 4.472135955 Solving 6 + a = 4.472135955 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + a = 4.472135955 + -6 Combine like terms: 6 + -6 = 0 0 + a = 4.472135955 + -6 a = 4.472135955 + -6 Combine like terms: 4.472135955 + -6 = -1.527864045 a = -1.527864045 Simplifying a = -1.527864045Subproblem 2
a + 6 = -4.472135955 Simplifying a + 6 = -4.472135955 Reorder the terms: 6 + a = -4.472135955 Solving 6 + a = -4.472135955 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + a = -4.472135955 + -6 Combine like terms: 6 + -6 = 0 0 + a = -4.472135955 + -6 a = -4.472135955 + -6 Combine like terms: -4.472135955 + -6 = -10.472135955 a = -10.472135955 Simplifying a = -10.472135955Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.527864045, -10.472135955}
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