3a(a-9)=6(a+5)

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Solution for 3a(a-9)=6(a+5) equation: 


Simplifying
3a(a + -9) = 6(a + 5)

Reorder the terms:
3a(-9 + a) = 6(a + 5)
(-9 * 3a + a * 3a) = 6(a + 5)
(-27a + 3a2) = 6(a + 5)

Reorder the terms:
-27a + 3a2 = 6(5 + a)
-27a + 3a2 = (5 * 6 + a * 6)
-27a + 3a2 = (30 + 6a)

Solving
-27a + 3a2 = 30 + 6a

Solving for variable 'a'.

Reorder the terms:
-30 + -27a + -6a + 3a2 = 30 + 6a + -30 + -6a

Combine like terms: -27a + -6a = -33a
-30 + -33a + 3a2 = 30 + 6a + -30 + -6a

Reorder the terms:
-30 + -33a + 3a2 = 30 + -30 + 6a + -6a

Combine like terms: 30 + -30 = 0
-30 + -33a + 3a2 = 0 + 6a + -6a
-30 + -33a + 3a2 = 6a + -6a

Combine like terms: 6a + -6a = 0
-30 + -33a + 3a2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-10 + -11a + a2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-10 + -11a + a2)' equal to zero and attempt to solve:

Simplifying
-10 + -11a + a2 = 0

Solving
-10 + -11a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -11a + 10 + a2 = 0 + 10

Reorder the terms:
-10 + 10 + -11a + a2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -11a + a2 = 0 + 10
-11a + a2 = 0 + 10

Combine like terms: 0 + 10 = 10
-11a + a2 = 10

The a term is -11a.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11a + 30.25 + a2 = 10 + 30.25

Reorder the terms:
30.25 + -11a + a2 = 10 + 30.25

Combine like terms: 10 + 30.25 = 40.25
30.25 + -11a + a2 = 40.25

Factor a perfect square on the left side:
(a + -5.5)(a + -5.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(a + -5.5) equal to 6.34428877 and -6.34428877.


Subproblem 1
a + -5.5 = 6.34428877

Simplifying
a + -5.5 = 6.34428877

Reorder the terms:
-5.5 + a = 6.34428877

Solving
-5.5 + a = 6.34428877

Solving for variable 'a'.

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

Add '5.5' to each side of the equation.
-5.5 + 5.5 + a = 6.34428877 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + a = 6.34428877 + 5.5
a = 6.34428877 + 5.5

Combine like terms: 6.34428877 + 5.5 = 11.84428877
a = 11.84428877

Simplifying
a = 11.84428877


Subproblem 2
a + -5.5 = -6.34428877

Simplifying
a + -5.5 = -6.34428877

Reorder the terms:
-5.5 + a = -6.34428877

Solving
-5.5 + a = -6.34428877

Solving for variable 'a'.

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

Add '5.5' to each side of the equation.
-5.5 + 5.5 + a = -6.34428877 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + a = -6.34428877 + 5.5
a = -6.34428877 + 5.5

Combine like terms: -6.34428877 + 5.5 = -0.84428877
a = -0.84428877

Simplifying
a = -0.84428877


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {11.84428877, -0.84428877}
Solution
a = {11.84428877, -0.84428877}

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