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Simplifying 0 = 12a2 + a + -4a3 Reorder the terms: 0 = a + 12a2 + -4a3 Solving 0 = a + 12a2 + -4a3 Solving for variable 'a'. Remove the zero: -1a + -12a2 + 4a3 = a + 12a2 + -4a3 + -1a + -12a2 + 4a3 Reorder the terms: -1a + -12a2 + 4a3 = a + -1a + 12a2 + -12a2 + -4a3 + 4a3 Combine like terms: a + -1a = 0 -1a + -12a2 + 4a3 = 0 + 12a2 + -12a2 + -4a3 + 4a3 -1a + -12a2 + 4a3 = 12a2 + -12a2 + -4a3 + 4a3 Combine like terms: 12a2 + -12a2 = 0 -1a + -12a2 + 4a3 = 0 + -4a3 + 4a3 -1a + -12a2 + 4a3 = -4a3 + 4a3 Combine like terms: -4a3 + 4a3 = 0 -1a + -12a2 + 4a3 = 0 Factor out the Greatest Common Factor (GCF), 'a'. a(-1 + -12a + 4a2) = 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 + -12a + 4a2)' equal to zero and attempt to solve: Simplifying -1 + -12a + 4a2 = 0 Solving -1 + -12a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -3a + a2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -3a + 0.25 + a2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -3a + a2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -3a + a2 = 0 + 0.25 -3a + a2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -3a + a2 = 0.25 The a term is -3a. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3a + 2.25 + a2 = 0.25 + 2.25 Reorder the terms: 2.25 + -3a + a2 = 0.25 + 2.25 Combine like terms: 0.25 + 2.25 = 2.5 2.25 + -3a + a2 = 2.5 Factor a perfect square on the left side: (a + -1.5)(a + -1.5) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (a + -1.5) equal to 1.58113883 and -1.58113883.Subproblem 1
a + -1.5 = 1.58113883 Simplifying a + -1.5 = 1.58113883 Reorder the terms: -1.5 + a = 1.58113883 Solving -1.5 + a = 1.58113883 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = 1.58113883 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = 1.58113883 + 1.5 a = 1.58113883 + 1.5 Combine like terms: 1.58113883 + 1.5 = 3.08113883 a = 3.08113883 Simplifying a = 3.08113883Subproblem 2
a + -1.5 = -1.58113883 Simplifying a + -1.5 = -1.58113883 Reorder the terms: -1.5 + a = -1.58113883 Solving -1.5 + a = -1.58113883 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = -1.58113883 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = -1.58113883 + 1.5 a = -1.58113883 + 1.5 Combine like terms: -1.58113883 + 1.5 = -0.08113883 a = -0.08113883 Simplifying a = -0.08113883Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.08113883, -0.08113883}Solution
a = {0, 3.08113883, -0.08113883}
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