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Simplifying 2 + 4a + 4 + a2 + 2a + 5a + 3 = 0 Reorder the terms: 2 + 4 + 3 + 4a + 2a + 5a + a2 = 0 Combine like terms: 2 + 4 = 6 6 + 3 + 4a + 2a + 5a + a2 = 0 Combine like terms: 6 + 3 = 9 9 + 4a + 2a + 5a + a2 = 0 Combine like terms: 4a + 2a = 6a 9 + 6a + 5a + a2 = 0 Combine like terms: 6a + 5a = 11a 9 + 11a + a2 = 0 Solving 9 + 11a + a2 = 0 Solving for variable 'a'. Begin completing the square. Move the constant term to the right: Add '-9' to each side of the equation. 9 + 11a + -9 + a2 = 0 + -9 Reorder the terms: 9 + -9 + 11a + a2 = 0 + -9 Combine like terms: 9 + -9 = 0 0 + 11a + a2 = 0 + -9 11a + a2 = 0 + -9 Combine like terms: 0 + -9 = -9 11a + a2 = -9 The a term is 11a. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11a + 30.25 + a2 = -9 + 30.25 Reorder the terms: 30.25 + 11a + a2 = -9 + 30.25 Combine like terms: -9 + 30.25 = 21.25 30.25 + 11a + a2 = 21.25 Factor a perfect square on the left side: (a + 5.5)(a + 5.5) = 21.25 Calculate the square root of the right side: 4.609772229 Break this problem into two subproblems by setting (a + 5.5) equal to 4.609772229 and -4.609772229.Subproblem 1
a + 5.5 = 4.609772229 Simplifying a + 5.5 = 4.609772229 Reorder the terms: 5.5 + a = 4.609772229 Solving 5.5 + a = 4.609772229 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + a = 4.609772229 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + a = 4.609772229 + -5.5 a = 4.609772229 + -5.5 Combine like terms: 4.609772229 + -5.5 = -0.890227771 a = -0.890227771 Simplifying a = -0.890227771Subproblem 2
a + 5.5 = -4.609772229 Simplifying a + 5.5 = -4.609772229 Reorder the terms: 5.5 + a = -4.609772229 Solving 5.5 + a = -4.609772229 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + a = -4.609772229 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + a = -4.609772229 + -5.5 a = -4.609772229 + -5.5 Combine like terms: -4.609772229 + -5.5 = -10.109772229 a = -10.109772229 Simplifying a = -10.109772229Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.890227771, -10.109772229}
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