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Simplifying 2a3 + a2 + -24a = 0 Reorder the terms: -24a + a2 + 2a3 = 0 Solving -24a + a2 + 2a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'a'. a(-24 + a + 2a2) = 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 '(-24 + a + 2a2)' equal to zero and attempt to solve: Simplifying -24 + a + 2a2 = 0 Solving -24 + a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -12 + 0.5a + a2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 0.5a + 12 + a2 = 0 + 12 Reorder the terms: -12 + 12 + 0.5a + a2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 0.5a + a2 = 0 + 12 0.5a + a2 = 0 + 12 Combine like terms: 0 + 12 = 12 0.5a + a2 = 12 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. 0.5a + 0.25 + a2 = 12 + 0.25 Reorder the terms: 0.25 + 0.5a + a2 = 12 + 0.25 Combine like terms: 12 + 0.25 = 12.25 0.25 + 0.5a + a2 = 12.25 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 12.25 Calculate the square root of the right side: 3.5 Break this problem into two subproblems by setting (a + 0.5) equal to 3.5 and -3.5.Subproblem 1
a + 0.5 = 3.5 Simplifying a + 0.5 = 3.5 Reorder the terms: 0.5 + a = 3.5 Solving 0.5 + a = 3.5 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 = 3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 3.5 + -0.5 a = 3.5 + -0.5 Combine like terms: 3.5 + -0.5 = 3 a = 3 Simplifying a = 3Subproblem 2
a + 0.5 = -3.5 Simplifying a + 0.5 = -3.5 Reorder the terms: 0.5 + a = -3.5 Solving 0.5 + a = -3.5 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 = -3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -3.5 + -0.5 a = -3.5 + -0.5 Combine like terms: -3.5 + -0.5 = -4 a = -4 Simplifying a = -4Solution
The solution to the problem is based on the solutions from the subproblems. a = {3, -4}Solution
a = {0, 3, -4}
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