ab=(a+3b)(a-3b)

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Solution for ab=(a+3b)(a-3b) equation: 


Simplifying
ab = (a + 3b)(a + -3b)

Multiply (a + 3b) * (a + -3b)
ab = (a(a + -3b) + 3b * (a + -3b))
ab = ((a * a + -3b * a) + 3b * (a + -3b))

Reorder the terms:
ab = ((-3ab + a2) + 3b * (a + -3b))
ab = ((-3ab + a2) + 3b * (a + -3b))
ab = (-3ab + a2 + (a * 3b + -3b * 3b))
ab = (-3ab + a2 + (3ab + -9b2))

Reorder the terms:
ab = (-3ab + 3ab + a2 + -9b2)

Combine like terms: -3ab + 3ab = 0
ab = (0 + a2 + -9b2)
ab = (a2 + -9b2)

Solving
ab = a2 + -9b2

Solving for variable 'a'.

Reorder the terms:
ab + -1a2 + 9b2 = a2 + -1a2 + -9b2 + 9b2

Combine like terms: a2 + -1a2 = 0
ab + -1a2 + 9b2 = 0 + -9b2 + 9b2
ab + -1a2 + 9b2 = -9b2 + 9b2

Combine like terms: -9b2 + 9b2 = 0
ab + -1a2 + 9b2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1ab + a2 + -9b2 = 0

Move the constant term to the right:

Add '9b2' to each side of the equation.
-1ab + a2 + -9b2 + 9b2 = 0 + 9b2

Combine like terms: -9b2 + 9b2 = 0
-1ab + a2 + 0 = 0 + 9b2
-1ab + a2 = 0 + 9b2
Remove the zero:
-1ab + a2 = 9b2

The a term is ab.  Take half its coefficient (0.5b).
Square it (0.25b2) and add it to both sides.

Add '0.25b2' to each side of the equation.
-1ab + a2 + 0.25b2 = 9b2 + 0.25b2

Combine like terms: 9b2 + 0.25b2 = 9.25b2
-1ab + a2 + 0.25b2 = 9.25b2

Factor a perfect square on the left side:
(a + 0.5b)(a + 0.5b) = 9.25b2

Calculate the square root of the right side: 3.041381265b

Break this problem into two subproblems by setting 
(a + 0.5b) equal to 3.041381265b and -3.041381265b.


Subproblem 1
a + 0.5b = 3.041381265b

Simplifying
a + 0.5b = 3.041381265b

Solving
a + 0.5b = 3.041381265b

Solving for variable 'a'.

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

Add '-0.5b' to each side of the equation.
a + 0.5b + -0.5b = 3.041381265b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = 3.041381265b + -0.5b
a = 3.041381265b + -0.5b

Combine like terms: 3.041381265b + -0.5b = 2.541381265b
a = 2.541381265b

Simplifying
a = 2.541381265b


Subproblem 2
a + 0.5b = -3.041381265b

Simplifying
a + 0.5b = -3.041381265b

Solving
a + 0.5b = -3.041381265b

Solving for variable 'a'.

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

Add '-0.5b' to each side of the equation.
a + 0.5b + -0.5b = -3.041381265b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = -3.041381265b + -0.5b
a = -3.041381265b + -0.5b

Combine like terms: -3.041381265b + -0.5b = -3.541381265b
a = -3.541381265b

Simplifying
a = -3.541381265b


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.541381265b, -3.541381265b}

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