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Simplifying 12a3 + -24a2 + -48a = 0 Reorder the terms: -48a + -24a2 + 12a3 = 0 Solving -48a + -24a2 + 12a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '12a'. 12a(-4 + -2a + a2) = 0 Ignore the factor 12.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 '(-4 + -2a + a2)' equal to zero and attempt to solve: Simplifying -4 + -2a + a2 = 0 Solving -4 + -2a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + -2a + 4 + a2 = 0 + 4 Reorder the terms: -4 + 4 + -2a + a2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -2a + a2 = 0 + 4 -2a + a2 = 0 + 4 Combine like terms: 0 + 4 = 4 -2a + a2 = 4 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 = 4 + 1 Reorder the terms: 1 + -2a + a2 = 4 + 1 Combine like terms: 4 + 1 = 5 1 + -2a + a2 = 5 Factor a perfect square on the left side: (a + -1)(a + -1) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (a + -1) equal to 2.236067978 and -2.236067978.Subproblem 1
a + -1 = 2.236067978 Simplifying a + -1 = 2.236067978 Reorder the terms: -1 + a = 2.236067978 Solving -1 + a = 2.236067978 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 = 2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + a = 2.236067978 + 1 a = 2.236067978 + 1 Combine like terms: 2.236067978 + 1 = 3.236067978 a = 3.236067978 Simplifying a = 3.236067978Subproblem 2
a + -1 = -2.236067978 Simplifying a + -1 = -2.236067978 Reorder the terms: -1 + a = -2.236067978 Solving -1 + a = -2.236067978 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 = -2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + a = -2.236067978 + 1 a = -2.236067978 + 1 Combine like terms: -2.236067978 + 1 = -1.236067978 a = -1.236067978 Simplifying a = -1.236067978Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.236067978, -1.236067978}Solution
a = {0, 3.236067978, -1.236067978}
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