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Simplifying 12a2 + 12a + -5 = 0 Reorder the terms: -5 + 12a + 12a2 = 0 Solving -5 + 12a + 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 + a + a2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + a + 0.4166666667 + a2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + a + a2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + a + a2 = 0 + 0.4166666667 a + a2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 a + a2 = 0.4166666667 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 = 0.4166666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + a2 = 0.4166666667 + 0.25 Combine like terms: 0.4166666667 + 0.25 = 0.6666666667 1.25 + a2 = 0.6666666667 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 0.6666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (a + 0.5) equal to 0.816496581 and -0.816496581.Subproblem 1
a + 0.5 = 0.816496581 Simplifying a + 0.5 = 0.816496581 Reorder the terms: 0.5 + a = 0.816496581 Solving 0.5 + a = 0.816496581 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 = 0.816496581 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.816496581 + -0.5 a = 0.816496581 + -0.5 Combine like terms: 0.816496581 + -0.5 = 0.316496581 a = 0.316496581 Simplifying a = 0.316496581Subproblem 2
a + 0.5 = -0.816496581 Simplifying a + 0.5 = -0.816496581 Reorder the terms: 0.5 + a = -0.816496581 Solving 0.5 + a = -0.816496581 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 = -0.816496581 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.816496581 + -0.5 a = -0.816496581 + -0.5 Combine like terms: -0.816496581 + -0.5 = -1.316496581 a = -1.316496581 Simplifying a = -1.316496581Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.316496581, -1.316496581}
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