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Simplifying (a + -6) + (2a + 5)(a + 6) = 0 Reorder the terms: (-6 + a) + (2a + 5)(a + 6) = 0 Remove parenthesis around (-6 + a) -6 + a + (2a + 5)(a + 6) = 0 Reorder the terms: -6 + a + (5 + 2a)(a + 6) = 0 Reorder the terms: -6 + a + (5 + 2a)(6 + a) = 0 Multiply (5 + 2a) * (6 + a) -6 + a + (5(6 + a) + 2a * (6 + a)) = 0 -6 + a + ((6 * 5 + a * 5) + 2a * (6 + a)) = 0 -6 + a + ((30 + 5a) + 2a * (6 + a)) = 0 -6 + a + (30 + 5a + (6 * 2a + a * 2a)) = 0 -6 + a + (30 + 5a + (12a + 2a2)) = 0 Combine like terms: 5a + 12a = 17a -6 + a + (30 + 17a + 2a2) = 0 Reorder the terms: -6 + 30 + a + 17a + 2a2 = 0 Combine like terms: -6 + 30 = 24 24 + a + 17a + 2a2 = 0 Combine like terms: a + 17a = 18a 24 + 18a + 2a2 = 0 Solving 24 + 18a + 2a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(12 + 9a + a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(12 + 9a + a2)' equal to zero and attempt to solve: Simplifying 12 + 9a + a2 = 0 Solving 12 + 9a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-12' to each side of the equation. 12 + 9a + -12 + a2 = 0 + -12 Reorder the terms: 12 + -12 + 9a + a2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + 9a + a2 = 0 + -12 9a + a2 = 0 + -12 Combine like terms: 0 + -12 = -12 9a + a2 = -12 The a term is 9a. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9a + 20.25 + a2 = -12 + 20.25 Reorder the terms: 20.25 + 9a + a2 = -12 + 20.25 Combine like terms: -12 + 20.25 = 8.25 20.25 + 9a + a2 = 8.25 Factor a perfect square on the left side: (a + 4.5)(a + 4.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (a + 4.5) equal to 2.872281323 and -2.872281323.Subproblem 1
a + 4.5 = 2.872281323 Simplifying a + 4.5 = 2.872281323 Reorder the terms: 4.5 + a = 2.872281323 Solving 4.5 + a = 2.872281323 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + a = 2.872281323 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = 2.872281323 + -4.5 a = 2.872281323 + -4.5 Combine like terms: 2.872281323 + -4.5 = -1.627718677 a = -1.627718677 Simplifying a = -1.627718677Subproblem 2
a + 4.5 = -2.872281323 Simplifying a + 4.5 = -2.872281323 Reorder the terms: 4.5 + a = -2.872281323 Solving 4.5 + a = -2.872281323 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + a = -2.872281323 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = -2.872281323 + -4.5 a = -2.872281323 + -4.5 Combine like terms: -2.872281323 + -4.5 = -7.372281323 a = -7.372281323 Simplifying a = -7.372281323Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.627718677, -7.372281323}Solution
a = {-1.627718677, -7.372281323}
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