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Simplifying 12a3 + -9a2 + -6a = 0 Reorder the terms: -6a + -9a2 + 12a3 = 0 Solving -6a + -9a2 + 12a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '3a'. 3a(-2 + -3a + 4a2) = 0 Ignore the factor 3.Subproblem 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 '(-2 + -3a + 4a2)' equal to zero and attempt to solve: Simplifying -2 + -3a + 4a2 = 0 Solving -2 + -3a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + -0.75a + a2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.75a + 0.5 + a2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.75a + a2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.75a + a2 = 0 + 0.5 -0.75a + a2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.75a + a2 = 0.5 The a term is -0.75a. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75a + 0.140625 + a2 = 0.5 + 0.140625 Reorder the terms: 0.140625 + -0.75a + a2 = 0.5 + 0.140625 Combine like terms: 0.5 + 0.140625 = 0.640625 0.140625 + -0.75a + a2 = 0.640625 Factor a perfect square on the left side: (a + -0.375)(a + -0.375) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (a + -0.375) equal to 0.80039053 and -0.80039053.Subproblem 1
a + -0.375 = 0.80039053 Simplifying a + -0.375 = 0.80039053 Reorder the terms: -0.375 + a = 0.80039053 Solving -0.375 + a = 0.80039053 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + a = 0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + a = 0.80039053 + 0.375 a = 0.80039053 + 0.375 Combine like terms: 0.80039053 + 0.375 = 1.17539053 a = 1.17539053 Simplifying a = 1.17539053Subproblem 2
a + -0.375 = -0.80039053 Simplifying a + -0.375 = -0.80039053 Reorder the terms: -0.375 + a = -0.80039053 Solving -0.375 + a = -0.80039053 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + a = -0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + a = -0.80039053 + 0.375 a = -0.80039053 + 0.375 Combine like terms: -0.80039053 + 0.375 = -0.42539053 a = -0.42539053 Simplifying a = -0.42539053Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.17539053, -0.42539053}Solution
a = {0, 1.17539053, -0.42539053}
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