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Simplifying 9a = -1a2 + -19 Reorder the terms: 9a = -19 + -1a2 Solving 9a = -19 + -1a2 Solving for variable 'a'. Reorder the terms: 19 + 9a + a2 = -19 + -1a2 + 19 + a2 Reorder the terms: 19 + 9a + a2 = -19 + 19 + -1a2 + a2 Combine like terms: -19 + 19 = 0 19 + 9a + a2 = 0 + -1a2 + a2 19 + 9a + a2 = -1a2 + a2 Combine like terms: -1a2 + a2 = 0 19 + 9a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-19' to each side of the equation. 19 + 9a + -19 + a2 = 0 + -19 Reorder the terms: 19 + -19 + 9a + a2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + 9a + a2 = 0 + -19 9a + a2 = 0 + -19 Combine like terms: 0 + -19 = -19 9a + a2 = -19 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 = -19 + 20.25 Reorder the terms: 20.25 + 9a + a2 = -19 + 20.25 Combine like terms: -19 + 20.25 = 1.25 20.25 + 9a + a2 = 1.25 Factor a perfect square on the left side: (a + 4.5)(a + 4.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (a + 4.5) equal to 1.118033989 and -1.118033989.Subproblem 1
a + 4.5 = 1.118033989 Simplifying a + 4.5 = 1.118033989 Reorder the terms: 4.5 + a = 1.118033989 Solving 4.5 + a = 1.118033989 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 = 1.118033989 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = 1.118033989 + -4.5 a = 1.118033989 + -4.5 Combine like terms: 1.118033989 + -4.5 = -3.381966011 a = -3.381966011 Simplifying a = -3.381966011Subproblem 2
a + 4.5 = -1.118033989 Simplifying a + 4.5 = -1.118033989 Reorder the terms: 4.5 + a = -1.118033989 Solving 4.5 + a = -1.118033989 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 = -1.118033989 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = -1.118033989 + -4.5 a = -1.118033989 + -4.5 Combine like terms: -1.118033989 + -4.5 = -5.618033989 a = -5.618033989 Simplifying a = -5.618033989Solution
The solution to the problem is based on the solutions from the subproblems. a = {-3.381966011, -5.618033989}
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