19a-3a(a-6)=9

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Solution for 19a-3a(a-6)=9 equation:


Simplifying
19a + -3a(a + -6) = 9

Reorder the terms:
19a + -3a(-6 + a) = 9
19a + (-6 * -3a + a * -3a) = 9
19a + (18a + -3a2) = 9

Combine like terms: 19a + 18a = 37a
37a + -3a2 = 9

Solving
37a + -3a2 = 9

Solving for variable 'a'.

Reorder the terms:
-9 + 37a + -3a2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 37a + -3a2 = 0

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

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

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -12.33333333a + -3 + a2 = 0 + -3

Reorder the terms:
3 + -3 + -12.33333333a + a2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -12.33333333a + a2 = 0 + -3
-12.33333333a + a2 = 0 + -3

Combine like terms: 0 + -3 = -3
-12.33333333a + a2 = -3

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 = -3 + 38.02777776

Reorder the terms:
38.02777776 + -12.33333333a + a2 = -3 + 38.02777776

Combine like terms: -3 + 38.02777776 = 35.02777776
38.02777776 + -12.33333333a + a2 = 35.02777776

Factor a perfect square on the left side:
(a + -6.166666665)(a + -6.166666665) = 35.02777776

Calculate the square root of the right side: 5.918426967

Break this problem into two subproblems by setting 
(a + -6.166666665) equal to 5.918426967 and -5.918426967.

Subproblem 1

a + -6.166666665 = 5.918426967 Simplifying a + -6.166666665 = 5.918426967 Reorder the terms: -6.166666665 + a = 5.918426967 Solving -6.166666665 + a = 5.918426967 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 = 5.918426967 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + a = 5.918426967 + 6.166666665 a = 5.918426967 + 6.166666665 Combine like terms: 5.918426967 + 6.166666665 = 12.085093632 a = 12.085093632 Simplifying a = 12.085093632

Subproblem 2

a + -6.166666665 = -5.918426967 Simplifying a + -6.166666665 = -5.918426967 Reorder the terms: -6.166666665 + a = -5.918426967 Solving -6.166666665 + a = -5.918426967 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 = -5.918426967 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + a = -5.918426967 + 6.166666665 a = -5.918426967 + 6.166666665 Combine like terms: -5.918426967 + 6.166666665 = 0.248239698 a = 0.248239698 Simplifying a = 0.248239698

Solution

The solution to the problem is based on the solutions from the subproblems. a = {12.085093632, 0.248239698}

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