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Simplifying 7a2 * 32 = 7 + -40a Reorder the terms for easier multiplication: 7 * 32a2 = 7 + -40a Multiply 7 * 32 224a2 = 7 + -40a Solving 224a2 = 7 + -40a Solving for variable 'a'. Reorder the terms: -7 + 40a + 224a2 = 7 + -40a + -7 + 40a Reorder the terms: -7 + 40a + 224a2 = 7 + -7 + -40a + 40a Combine like terms: 7 + -7 = 0 -7 + 40a + 224a2 = 0 + -40a + 40a -7 + 40a + 224a2 = -40a + 40a Combine like terms: -40a + 40a = 0 -7 + 40a + 224a2 = 0 Begin completing the square. Divide all terms by 224 the coefficient of the squared term: Divide each side by '224'. -0.03125 + 0.1785714286a + a2 = 0 Move the constant term to the right: Add '0.03125' to each side of the equation. -0.03125 + 0.1785714286a + 0.03125 + a2 = 0 + 0.03125 Reorder the terms: -0.03125 + 0.03125 + 0.1785714286a + a2 = 0 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + 0.1785714286a + a2 = 0 + 0.03125 0.1785714286a + a2 = 0 + 0.03125 Combine like terms: 0 + 0.03125 = 0.03125 0.1785714286a + a2 = 0.03125 The a term is 0.1785714286a. Take half its coefficient (0.0892857143). Square it (0.007971938778) and add it to both sides. Add '0.007971938778' to each side of the equation. 0.1785714286a + 0.007971938778 + a2 = 0.03125 + 0.007971938778 Reorder the terms: 0.007971938778 + 0.1785714286a + a2 = 0.03125 + 0.007971938778 Combine like terms: 0.03125 + 0.007971938778 = 0.039221938778 0.007971938778 + 0.1785714286a + a2 = 0.039221938778 Factor a perfect square on the left side: (a + 0.0892857143)(a + 0.0892857143) = 0.039221938778 Calculate the square root of the right side: 0.198045295 Break this problem into two subproblems by setting (a + 0.0892857143) equal to 0.198045295 and -0.198045295.Subproblem 1
a + 0.0892857143 = 0.198045295 Simplifying a + 0.0892857143 = 0.198045295 Reorder the terms: 0.0892857143 + a = 0.198045295 Solving 0.0892857143 + a = 0.198045295 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.0892857143' to each side of the equation. 0.0892857143 + -0.0892857143 + a = 0.198045295 + -0.0892857143 Combine like terms: 0.0892857143 + -0.0892857143 = 0.0000000000 0.0000000000 + a = 0.198045295 + -0.0892857143 a = 0.198045295 + -0.0892857143 Combine like terms: 0.198045295 + -0.0892857143 = 0.1087595807 a = 0.1087595807 Simplifying a = 0.1087595807Subproblem 2
a + 0.0892857143 = -0.198045295 Simplifying a + 0.0892857143 = -0.198045295 Reorder the terms: 0.0892857143 + a = -0.198045295 Solving 0.0892857143 + a = -0.198045295 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.0892857143' to each side of the equation. 0.0892857143 + -0.0892857143 + a = -0.198045295 + -0.0892857143 Combine like terms: 0.0892857143 + -0.0892857143 = 0.0000000000 0.0000000000 + a = -0.198045295 + -0.0892857143 a = -0.198045295 + -0.0892857143 Combine like terms: -0.198045295 + -0.0892857143 = -0.2873310093 a = -0.2873310093 Simplifying a = -0.2873310093Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.1087595807, -0.2873310093}
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