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Simplifying a = a * a(3 + -1a) Multiply a * a a = a2(3 + -1a) a = (3 * a2 + -1a * a2) a = (3a2 + -1a3) Solving a = 3a2 + -1a3 Solving for variable 'a'. Reorder the terms: a + -3a2 + a3 = 3a2 + -3a2 + -1a3 + a3 Combine like terms: 3a2 + -3a2 = 0 a + -3a2 + a3 = 0 + -1a3 + a3 a + -3a2 + a3 = -1a3 + a3 Combine like terms: -1a3 + a3 = 0 a + -3a2 + a3 = 0 Factor out the Greatest Common Factor (GCF), 'a'. a(1 + -3a + a2) = 0Subproblem 1
Set the factor 'a' equal to zero and attempt to solve: Simplifying a = 0 Solving a = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a = 0Subproblem 2
Set the factor '(1 + -3a + a2)' equal to zero and attempt to solve: Simplifying 1 + -3a + a2 = 0 Solving 1 + -3a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3a + -1 + a2 = 0 + -1 Reorder the terms: 1 + -1 + -3a + a2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3a + a2 = 0 + -1 -3a + a2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3a + a2 = -1 The a term is -3a. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3a + 2.25 + a2 = -1 + 2.25 Reorder the terms: 2.25 + -3a + a2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3a + a2 = 1.25 Factor a perfect square on the left side: (a + -1.5)(a + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (a + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
a + -1.5 = 1.118033989 Simplifying a + -1.5 = 1.118033989 Reorder the terms: -1.5 + a = 1.118033989 Solving -1.5 + a = 1.118033989 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = 1.118033989 + 1.5 a = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 a = 2.618033989 Simplifying a = 2.618033989Subproblem 2
a + -1.5 = -1.118033989 Simplifying a + -1.5 = -1.118033989 Reorder the terms: -1.5 + a = -1.118033989 Solving -1.5 + a = -1.118033989 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = -1.118033989 + 1.5 a = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 a = 0.381966011 Simplifying a = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.618033989, 0.381966011}Solution
a = {0, 2.618033989, 0.381966011}
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