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Simplifying 4a3 + 2a2 + -23a = 0 Reorder the terms: -23a + 2a2 + 4a3 = 0 Solving -23a + 2a2 + 4a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'a'. a(-23 + 2a + 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 '(-23 + 2a + 4a2)' equal to zero and attempt to solve: Simplifying -23 + 2a + 4a2 = 0 Solving -23 + 2a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5.75 + 0.5a + a2 = 0 Move the constant term to the right: Add '5.75' to each side of the equation. -5.75 + 0.5a + 5.75 + a2 = 0 + 5.75 Reorder the terms: -5.75 + 5.75 + 0.5a + a2 = 0 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + 0.5a + a2 = 0 + 5.75 0.5a + a2 = 0 + 5.75 Combine like terms: 0 + 5.75 = 5.75 0.5a + a2 = 5.75 The a term is 0.5a. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5a + 0.0625 + a2 = 5.75 + 0.0625 Reorder the terms: 0.0625 + 0.5a + a2 = 5.75 + 0.0625 Combine like terms: 5.75 + 0.0625 = 5.8125 0.0625 + 0.5a + a2 = 5.8125 Factor a perfect square on the left side: (a + 0.25)(a + 0.25) = 5.8125 Calculate the square root of the right side: 2.41091269 Break this problem into two subproblems by setting (a + 0.25) equal to 2.41091269 and -2.41091269.Subproblem 1
a + 0.25 = 2.41091269 Simplifying a + 0.25 = 2.41091269 Reorder the terms: 0.25 + a = 2.41091269 Solving 0.25 + a = 2.41091269 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = 2.41091269 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = 2.41091269 + -0.25 a = 2.41091269 + -0.25 Combine like terms: 2.41091269 + -0.25 = 2.16091269 a = 2.16091269 Simplifying a = 2.16091269Subproblem 2
a + 0.25 = -2.41091269 Simplifying a + 0.25 = -2.41091269 Reorder the terms: 0.25 + a = -2.41091269 Solving 0.25 + a = -2.41091269 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = -2.41091269 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = -2.41091269 + -0.25 a = -2.41091269 + -0.25 Combine like terms: -2.41091269 + -0.25 = -2.66091269 a = -2.66091269 Simplifying a = -2.66091269Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.16091269, -2.66091269}Solution
a = {0, 2.16091269, -2.66091269}
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