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Simplifying 12a2 + 12a + -6 = 0 Reorder the terms: -6 + 12a + 12a2 = 0 Solving -6 + 12a + 12a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '6'. 6(-1 + 2a + 2a2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-1 + 2a + 2a2)' equal to zero and attempt to solve: Simplifying -1 + 2a + 2a2 = 0 Solving -1 + 2a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + a + a2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + a + 0.5 + a2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + a + a2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a + a2 = 0 + 0.5 a + a2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 a + a2 = 0.5 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + a2 = 0.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + a2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 1.25 + a2 = 0.75 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (a + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
a + 0.5 = 0.866025404 Simplifying a + 0.5 = 0.866025404 Reorder the terms: 0.5 + a = 0.866025404 Solving 0.5 + a = 0.866025404 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.866025404 + -0.5 a = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 a = 0.366025404 Simplifying a = 0.366025404Subproblem 2
a + 0.5 = -0.866025404 Simplifying a + 0.5 = -0.866025404 Reorder the terms: 0.5 + a = -0.866025404 Solving 0.5 + a = -0.866025404 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.866025404 + -0.5 a = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 a = -1.366025404 Simplifying a = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.366025404, -1.366025404}Solution
a = {0.366025404, -1.366025404}
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