6a^2+ab-6b^2=0

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Solution for 6a^2+ab-6b^2=0 equation: 


Simplifying
6a2 + ab + -6b2 = 0

Reorder the terms:
ab + 6a2 + -6b2 = 0

Solving
ab + 6a2 + -6b2 = 0

Solving for variable 'a'.

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

Divide each side by '6'.
0.1666666667ab + a2 + -1b2 = 0

Move the constant term to the right:

Add 'b2' to each side of the equation.
0.1666666667ab + a2 + -1b2 + b2 = 0 + b2

Combine like terms: -1b2 + b2 = 0
0.1666666667ab + a2 + 0 = 0 + b2
0.1666666667ab + a2 = 0 + b2
Remove the zero:
0.1666666667ab + a2 = b2

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.
0.1666666667ab + a2 + 0.25b2 = b2 + 0.25b2

Combine like terms: b2 + 0.25b2 = 1.25b2
0.1666666667ab + a2 + 0.25b2 = 1.25b2

Factor a perfect square on the left side:
(a + 0.5b)(a + 0.5b) = 1.25b2

Calculate the square root of the right side: 1.118033989b

Break this problem into two subproblems by setting 
(a + 0.5b) equal to 1.118033989b and -1.118033989b.


Subproblem 1
a + 0.5b = 1.118033989b

Simplifying
a + 0.5b = 1.118033989b

Solving
a + 0.5b = 1.118033989b

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 = 1.118033989b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = 1.118033989b + -0.5b
a = 1.118033989b + -0.5b

Combine like terms: 1.118033989b + -0.5b = 0.618033989b
a = 0.618033989b

Simplifying
a = 0.618033989b


Subproblem 2
a + 0.5b = -1.118033989b

Simplifying
a + 0.5b = -1.118033989b

Solving
a + 0.5b = -1.118033989b

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 = -1.118033989b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = -1.118033989b + -0.5b
a = -1.118033989b + -0.5b

Combine like terms: -1.118033989b + -0.5b = -1.618033989b
a = -1.618033989b

Simplifying
a = -1.618033989b


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.618033989b, -1.618033989b}

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