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Simplifying 12a2 + -1a + -27 = 0 Reorder the terms: -27 + -1a + 12a2 = 0 Solving -27 + -1a + 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'. -2.25 + -0.08333333333a + a2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + -0.08333333333a + 2.25 + a2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + -0.08333333333a + a2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + -0.08333333333a + a2 = 0 + 2.25 -0.08333333333a + a2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 -0.08333333333a + a2 = 2.25 The a term is -0.08333333333a. Take half its coefficient (-0.04166666667). Square it (0.001736111111) and add it to both sides. Add '0.001736111111' to each side of the equation. -0.08333333333a + 0.001736111111 + a2 = 2.25 + 0.001736111111 Reorder the terms: 0.001736111111 + -0.08333333333a + a2 = 2.25 + 0.001736111111 Combine like terms: 2.25 + 0.001736111111 = 2.251736111111 0.001736111111 + -0.08333333333a + a2 = 2.251736111111 Factor a perfect square on the left side: (a + -0.04166666667)(a + -0.04166666667) = 2.251736111111 Calculate the square root of the right side: 1.500578592 Break this problem into two subproblems by setting (a + -0.04166666667) equal to 1.500578592 and -1.500578592.Subproblem 1
a + -0.04166666667 = 1.500578592 Simplifying a + -0.04166666667 = 1.500578592 Reorder the terms: -0.04166666667 + a = 1.500578592 Solving -0.04166666667 + a = 1.500578592 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.04166666667' to each side of the equation. -0.04166666667 + 0.04166666667 + a = 1.500578592 + 0.04166666667 Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000 0.00000000000 + a = 1.500578592 + 0.04166666667 a = 1.500578592 + 0.04166666667 Combine like terms: 1.500578592 + 0.04166666667 = 1.54224525867 a = 1.54224525867 Simplifying a = 1.54224525867Subproblem 2
a + -0.04166666667 = -1.500578592 Simplifying a + -0.04166666667 = -1.500578592 Reorder the terms: -0.04166666667 + a = -1.500578592 Solving -0.04166666667 + a = -1.500578592 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.04166666667' to each side of the equation. -0.04166666667 + 0.04166666667 + a = -1.500578592 + 0.04166666667 Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000 0.00000000000 + a = -1.500578592 + 0.04166666667 a = -1.500578592 + 0.04166666667 Combine like terms: -1.500578592 + 0.04166666667 = -1.45891192533 a = -1.45891192533 Simplifying a = -1.45891192533Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.54224525867, -1.45891192533}
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