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Simplifying 12a3 + -24a2 + 4a = 0 Reorder the terms: 4a + -24a2 + 12a3 = 0 Solving 4a + -24a2 + 12a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '4a'. 4a(1 + -6a + 3a2) = 0 Ignore the factor 4.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 '(1 + -6a + 3a2)' equal to zero and attempt to solve: Simplifying 1 + -6a + 3a2 = 0 Solving 1 + -6a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + -2a + a2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -2a + -0.3333333333 + a2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -2a + a2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -2a + a2 = 0 + -0.3333333333 -2a + a2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -2a + a2 = -0.3333333333 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 = -0.3333333333 + 1 Reorder the terms: 1 + -2a + a2 = -0.3333333333 + 1 Combine like terms: -0.3333333333 + 1 = 0.6666666667 1 + -2a + a2 = 0.6666666667 Factor a perfect square on the left side: (a + -1)(a + -1) = 0.6666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (a + -1) equal to 0.816496581 and -0.816496581.Subproblem 1
a + -1 = 0.816496581 Simplifying a + -1 = 0.816496581 Reorder the terms: -1 + a = 0.816496581 Solving -1 + a = 0.816496581 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 = 0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + a = 0.816496581 + 1 a = 0.816496581 + 1 Combine like terms: 0.816496581 + 1 = 1.816496581 a = 1.816496581 Simplifying a = 1.816496581Subproblem 2
a + -1 = -0.816496581 Simplifying a + -1 = -0.816496581 Reorder the terms: -1 + a = -0.816496581 Solving -1 + a = -0.816496581 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 = -0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + a = -0.816496581 + 1 a = -0.816496581 + 1 Combine like terms: -0.816496581 + 1 = 0.183503419 a = 0.183503419 Simplifying a = 0.183503419Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.816496581, 0.183503419}Solution
a = {0, 1.816496581, 0.183503419}
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