19a-3a(a-6)=66

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Solution for 19a-3a(a-6)=66 equation:


Simplifying
19a + -3a(a + -6) = 66

Reorder the terms:
19a + -3a(-6 + a) = 66
19a + (-6 * -3a + a * -3a) = 66
19a + (18a + -3a2) = 66

Combine like terms: 19a + 18a = 37a
37a + -3a2 = 66

Solving
37a + -3a2 = 66

Solving for variable 'a'.

Reorder the terms:
-66 + 37a + -3a2 = 66 + -66

Combine like terms: 66 + -66 = 0
-66 + 37a + -3a2 = 0

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

Divide each side by '-3'.
22 + -12.33333333a + a2 = 0

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + -12.33333333a + -22 + a2 = 0 + -22

Reorder the terms:
22 + -22 + -12.33333333a + a2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + -12.33333333a + a2 = 0 + -22
-12.33333333a + a2 = 0 + -22

Combine like terms: 0 + -22 = -22
-12.33333333a + a2 = -22

The a term is -12.33333333a.  Take half its coefficient (-6.166666665).
Square it (38.02777776) and add it to both sides.

Add '38.02777776' to each side of the equation.
-12.33333333a + 38.02777776 + a2 = -22 + 38.02777776

Reorder the terms:
38.02777776 + -12.33333333a + a2 = -22 + 38.02777776

Combine like terms: -22 + 38.02777776 = 16.02777776
38.02777776 + -12.33333333a + a2 = 16.02777776

Factor a perfect square on the left side:
(a + -6.166666665)(a + -6.166666665) = 16.02777776

Calculate the square root of the right side: 4.003470714

Break this problem into two subproblems by setting 
(a + -6.166666665) equal to 4.003470714 and -4.003470714.


Subproblem 1
a + -6.166666665 = 4.003470714

Simplifying
a + -6.166666665 = 4.003470714

Reorder the terms:
-6.166666665 + a = 4.003470714

Solving
-6.166666665 + a = 4.003470714

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '6.166666665' to each side of the equation.
-6.166666665 + 6.166666665 + a = 4.003470714 + 6.166666665

Combine like terms: -6.166666665 + 6.166666665 = 0.000000000
0.000000000 + a = 4.003470714 + 6.166666665
a = 4.003470714 + 6.166666665

Combine like terms: 4.003470714 + 6.166666665 = 10.170137379
a = 10.170137379

Simplifying
a = 10.170137379


Subproblem 2
a + -6.166666665 = -4.003470714

Simplifying
a + -6.166666665 = -4.003470714

Reorder the terms:
-6.166666665 + a = -4.003470714

Solving
-6.166666665 + a = -4.003470714

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '6.166666665' to each side of the equation.
-6.166666665 + 6.166666665 + a = -4.003470714 + 6.166666665

Combine like terms: -6.166666665 + 6.166666665 = 0.000000000
0.000000000 + a = -4.003470714 + 6.166666665
a = -4.003470714 + 6.166666665

Combine like terms: -4.003470714 + 6.166666665 = 2.163195951
a = 2.163195951

Simplifying
a = 2.163195951


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {10.170137379, 2.163195951}

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