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Simplifying 4a2 + 2a + -456 = 0 Reorder the terms: -456 + 2a + 4a2 = 0 Solving -456 + 2a + 4a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(-228 + a + 2a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-228 + a + 2a2)' equal to zero and attempt to solve: Simplifying -228 + a + 2a2 = 0 Solving -228 + a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -114 + 0.5a + a2 = 0 Move the constant term to the right: Add '114' to each side of the equation. -114 + 0.5a + 114 + a2 = 0 + 114 Reorder the terms: -114 + 114 + 0.5a + a2 = 0 + 114 Combine like terms: -114 + 114 = 0 0 + 0.5a + a2 = 0 + 114 0.5a + a2 = 0 + 114 Combine like terms: 0 + 114 = 114 0.5a + a2 = 114 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 = 114 + 0.25 Reorder the terms: 0.25 + 0.5a + a2 = 114 + 0.25 Combine like terms: 114 + 0.25 = 114.25 0.25 + 0.5a + a2 = 114.25 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 114.25 Calculate the square root of the right side: 10.688779163 Break this problem into two subproblems by setting (a + 0.5) equal to 10.688779163 and -10.688779163.Subproblem 1
a + 0.5 = 10.688779163 Simplifying a + 0.5 = 10.688779163 Reorder the terms: 0.5 + a = 10.688779163 Solving 0.5 + a = 10.688779163 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 = 10.688779163 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 10.688779163 + -0.5 a = 10.688779163 + -0.5 Combine like terms: 10.688779163 + -0.5 = 10.188779163 a = 10.188779163 Simplifying a = 10.188779163Subproblem 2
a + 0.5 = -10.688779163 Simplifying a + 0.5 = -10.688779163 Reorder the terms: 0.5 + a = -10.688779163 Solving 0.5 + a = -10.688779163 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 = -10.688779163 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -10.688779163 + -0.5 a = -10.688779163 + -0.5 Combine like terms: -10.688779163 + -0.5 = -11.188779163 a = -11.188779163 Simplifying a = -11.188779163Solution
The solution to the problem is based on the solutions from the subproblems. a = {10.188779163, -11.188779163}Solution
a = {10.188779163, -11.188779163}
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