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Simplifying 32a2 + 24a = 1 Reorder the terms: 24a + 32a2 = 1 Solving 24a + 32a2 = 1 Solving for variable 'a'. Reorder the terms: -1 + 24a + 32a2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 24a + 32a2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.03125 + 0.75a + a2 = 0 Move the constant term to the right: Add '0.03125' to each side of the equation. -0.03125 + 0.75a + 0.03125 + a2 = 0 + 0.03125 Reorder the terms: -0.03125 + 0.03125 + 0.75a + a2 = 0 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + 0.75a + a2 = 0 + 0.03125 0.75a + a2 = 0 + 0.03125 Combine like terms: 0 + 0.03125 = 0.03125 0.75a + a2 = 0.03125 The a term is 0.75a. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75a + 0.140625 + a2 = 0.03125 + 0.140625 Reorder the terms: 0.140625 + 0.75a + a2 = 0.03125 + 0.140625 Combine like terms: 0.03125 + 0.140625 = 0.171875 0.140625 + 0.75a + a2 = 0.171875 Factor a perfect square on the left side: (a + 0.375)(a + 0.375) = 0.171875 Calculate the square root of the right side: 0.414578099 Break this problem into two subproblems by setting (a + 0.375) equal to 0.414578099 and -0.414578099.Subproblem 1
a + 0.375 = 0.414578099 Simplifying a + 0.375 = 0.414578099 Reorder the terms: 0.375 + a = 0.414578099 Solving 0.375 + a = 0.414578099 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + a = 0.414578099 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + a = 0.414578099 + -0.375 a = 0.414578099 + -0.375 Combine like terms: 0.414578099 + -0.375 = 0.039578099 a = 0.039578099 Simplifying a = 0.039578099Subproblem 2
a + 0.375 = -0.414578099 Simplifying a + 0.375 = -0.414578099 Reorder the terms: 0.375 + a = -0.414578099 Solving 0.375 + a = -0.414578099 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + a = -0.414578099 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + a = -0.414578099 + -0.375 a = -0.414578099 + -0.375 Combine like terms: -0.414578099 + -0.375 = -0.789578099 a = -0.789578099 Simplifying a = -0.789578099Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.039578099, -0.789578099}
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