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Simplifying 12a2 + -6a + -5 = 0 Reorder the terms: -5 + -6a + 12a2 = 0 Solving -5 + -6a + 12a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.4166666667 + -0.5a + a2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + -0.5a + 0.4166666667 + a2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + -0.5a + a2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + -0.5a + a2 = 0 + 0.4166666667 -0.5a + a2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 -0.5a + a2 = 0.4166666667 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 = 0.4166666667 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 0.4166666667 + 0.0625 Combine like terms: 0.4166666667 + 0.0625 = 0.4791666667 0.0625 + -0.5a + a2 = 0.4791666667 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 0.4791666667 Calculate the square root of the right side: 0.692218655 Break this problem into two subproblems by setting (a + -0.25) equal to 0.692218655 and -0.692218655.Subproblem 1
a + -0.25 = 0.692218655 Simplifying a + -0.25 = 0.692218655 Reorder the terms: -0.25 + a = 0.692218655 Solving -0.25 + a = 0.692218655 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 = 0.692218655 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 0.692218655 + 0.25 a = 0.692218655 + 0.25 Combine like terms: 0.692218655 + 0.25 = 0.942218655 a = 0.942218655 Simplifying a = 0.942218655Subproblem 2
a + -0.25 = -0.692218655 Simplifying a + -0.25 = -0.692218655 Reorder the terms: -0.25 + a = -0.692218655 Solving -0.25 + a = -0.692218655 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 = -0.692218655 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -0.692218655 + 0.25 a = -0.692218655 + 0.25 Combine like terms: -0.692218655 + 0.25 = -0.442218655 a = -0.442218655 Simplifying a = -0.442218655Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.942218655, -0.442218655}
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