3a^2-23-12a=0

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Solution for 3a^2-23-12a=0 equation:


Simplifying
3a2 + -23 + -12a = 0

Reorder the terms:
-23 + -12a + 3a2 = 0

Solving
-23 + -12a + 3a2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-7.666666667 + -4a + a2 = 0

Move the constant term to the right:

Add '7.666666667' to each side of the equation.
-7.666666667 + -4a + 7.666666667 + a2 = 0 + 7.666666667

Reorder the terms:
-7.666666667 + 7.666666667 + -4a + a2 = 0 + 7.666666667

Combine like terms: -7.666666667 + 7.666666667 = 0.000000000
0.000000000 + -4a + a2 = 0 + 7.666666667
-4a + a2 = 0 + 7.666666667

Combine like terms: 0 + 7.666666667 = 7.666666667
-4a + a2 = 7.666666667

The a term is -4a.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4a + 4 + a2 = 7.666666667 + 4

Reorder the terms:
4 + -4a + a2 = 7.666666667 + 4

Combine like terms: 7.666666667 + 4 = 11.666666667
4 + -4a + a2 = 11.666666667

Factor a perfect square on the left side:
(a + -2)(a + -2) = 11.666666667

Calculate the square root of the right side: 3.415650255

Break this problem into two subproblems by setting 
(a + -2) equal to 3.415650255 and -3.415650255.

Subproblem 1

a + -2 = 3.415650255 Simplifying a + -2 = 3.415650255 Reorder the terms: -2 + a = 3.415650255 Solving -2 + a = 3.415650255 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 3.415650255 + 2 Combine like terms: -2 + 2 = 0 0 + a = 3.415650255 + 2 a = 3.415650255 + 2 Combine like terms: 3.415650255 + 2 = 5.415650255 a = 5.415650255 Simplifying a = 5.415650255

Subproblem 2

a + -2 = -3.415650255 Simplifying a + -2 = -3.415650255 Reorder the terms: -2 + a = -3.415650255 Solving -2 + a = -3.415650255 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -3.415650255 + 2 Combine like terms: -2 + 2 = 0 0 + a = -3.415650255 + 2 a = -3.415650255 + 2 Combine like terms: -3.415650255 + 2 = -1.415650255 a = -1.415650255 Simplifying a = -1.415650255

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.415650255, -1.415650255}

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