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Simplifying 32a2 + 32a + 4 = 0 Reorder the terms: 4 + 32a + 32a2 = 0 Solving 4 + 32a + 32a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '4'. 4(1 + 8a + 8a2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1 + 8a + 8a2)' equal to zero and attempt to solve: Simplifying 1 + 8a + 8a2 = 0 Solving 1 + 8a + 8a2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.125 + a + a2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + a + -0.125 + a2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + a + a2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a + a2 = 0 + -0.125 a + a2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 a + a2 = -0.125 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. + 0.25 + a2 = -0.125 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + a2 = -0.125 + 0.25 Combine like terms: -0.125 + 0.25 = 0.125 1.25 + a2 = 0.125 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (a + 0.5) equal to 0.353553391 and -0.353553391.Subproblem 1
a + 0.5 = 0.353553391 Simplifying a + 0.5 = 0.353553391 Reorder the terms: 0.5 + a = 0.353553391 Solving 0.5 + a = 0.353553391 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 = 0.353553391 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.353553391 + -0.5 a = 0.353553391 + -0.5 Combine like terms: 0.353553391 + -0.5 = -0.146446609 a = -0.146446609 Simplifying a = -0.146446609Subproblem 2
a + 0.5 = -0.353553391 Simplifying a + 0.5 = -0.353553391 Reorder the terms: 0.5 + a = -0.353553391 Solving 0.5 + a = -0.353553391 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 = -0.353553391 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.353553391 + -0.5 a = -0.353553391 + -0.5 Combine like terms: -0.353553391 + -0.5 = -0.853553391 a = -0.853553391 Simplifying a = -0.853553391Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.146446609, -0.853553391}Solution
a = {-0.146446609, -0.853553391}
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