-12a^2+23a+20=0

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Solution for -12a^2+23a+20=0 equation:


Simplifying
-12a2 + 23a + 20 = 0

Reorder the terms:
20 + 23a + -12a2 = 0

Solving
20 + 23a + -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'.
-1.666666667 + -1.916666667a + a2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -1.916666667a + 1.666666667 + a2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -1.916666667a + a2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -1.916666667a + a2 = 0 + 1.666666667
-1.916666667a + a2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-1.916666667a + a2 = 1.666666667

The a term is -1.916666667a.  Take half its coefficient (-0.9583333335).
Square it (0.9184027781) and add it to both sides.

Add '0.9184027781' to each side of the equation.
-1.916666667a + 0.9184027781 + a2 = 1.666666667 + 0.9184027781

Reorder the terms:
0.9184027781 + -1.916666667a + a2 = 1.666666667 + 0.9184027781

Combine like terms: 1.666666667 + 0.9184027781 = 2.5850694451
0.9184027781 + -1.916666667a + a2 = 2.5850694451

Factor a perfect square on the left side:
(a + -0.9583333335)(a + -0.9583333335) = 2.5850694451

Calculate the square root of the right side: 1.607815115

Break this problem into two subproblems by setting 
(a + -0.9583333335) equal to 1.607815115 and -1.607815115.

Subproblem 1

a + -0.9583333335 = 1.607815115 Simplifying a + -0.9583333335 = 1.607815115 Reorder the terms: -0.9583333335 + a = 1.607815115 Solving -0.9583333335 + a = 1.607815115 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9583333335' to each side of the equation. -0.9583333335 + 0.9583333335 + a = 1.607815115 + 0.9583333335 Combine like terms: -0.9583333335 + 0.9583333335 = 0.0000000000 0.0000000000 + a = 1.607815115 + 0.9583333335 a = 1.607815115 + 0.9583333335 Combine like terms: 1.607815115 + 0.9583333335 = 2.5661484485 a = 2.5661484485 Simplifying a = 2.5661484485

Subproblem 2

a + -0.9583333335 = -1.607815115 Simplifying a + -0.9583333335 = -1.607815115 Reorder the terms: -0.9583333335 + a = -1.607815115 Solving -0.9583333335 + a = -1.607815115 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9583333335' to each side of the equation. -0.9583333335 + 0.9583333335 + a = -1.607815115 + 0.9583333335 Combine like terms: -0.9583333335 + 0.9583333335 = 0.0000000000 0.0000000000 + a = -1.607815115 + 0.9583333335 a = -1.607815115 + 0.9583333335 Combine like terms: -1.607815115 + 0.9583333335 = -0.6494817815 a = -0.6494817815 Simplifying a = -0.6494817815

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.5661484485, -0.6494817815}

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