12a^2-a-27=0

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Solution for 12a^2-a-27=0 equation:


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 = 1.54224525867

Subproblem 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 = -1.45891192533

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.54224525867, -1.45891192533}

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