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Simplifying a3 + -2a2 + -1a = 0 Reorder the terms: -1a + -2a2 + a3 = 0 Solving -1a + -2a2 + a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'a'. a(-1 + -2a + 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 + -2a + a2)' equal to zero and attempt to solve: Simplifying -1 + -2a + a2 = 0 Solving -1 + -2a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -2a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + -2a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2a + a2 = 0 + 1 -2a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2a + a2 = 1 The a term is -2a. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2a + 1 + a2 = 1 + 1 Reorder the terms: 1 + -2a + a2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + -2a + a2 = 2 Factor a perfect square on the left side: (a + -1)(a + -1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (a + -1) equal to 1.414213562 and -1.414213562.Subproblem 1
a + -1 = 1.414213562 Simplifying a + -1 = 1.414213562 Reorder the terms: -1 + a = 1.414213562 Solving -1 + a = 1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + a = 1.414213562 + 1 a = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 a = 2.414213562 Simplifying a = 2.414213562Subproblem 2
a + -1 = -1.414213562 Simplifying a + -1 = -1.414213562 Reorder the terms: -1 + a = -1.414213562 Solving -1 + a = -1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + a = -1.414213562 + 1 a = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 a = -0.414213562 Simplifying a = -0.414213562Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.414213562, -0.414213562}Solution
a = {0, 2.414213562, -0.414213562}
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