3a^2+ab=b^2

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Solution for 3a^2+ab=b^2 equation:


Simplifying
3a2 + ab = b2

Reorder the terms:
ab + 3a2 = b2

Solving
ab + 3a2 = b2

Solving for variable 'a'.

Combine like terms: b2 + -1b2 = 0
ab + 3a2 + -1b2 = 0

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

Divide each side by '3'.
0.3333333333ab + a2 + -0.3333333333b2 = 0

Move the constant term to the right:

Add '0.3333333333b2' to each side of the equation.
0.3333333333ab + a2 + -0.3333333333b2 + 0.3333333333b2 = 0 + 0.3333333333b2

Combine like terms: -0.3333333333b2 + 0.3333333333b2 = 0.0000000000
0.3333333333ab + a2 + 0.0000000000 = 0 + 0.3333333333b2
0.3333333333ab + a2 = 0 + 0.3333333333b2
Remove the zero:
0.3333333333ab + a2 = 0.3333333333b2

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.
0.3333333333ab + a2 + 0.25b2 = 0.3333333333b2 + 0.25b2

Combine like terms: 0.3333333333b2 + 0.25b2 = 0.5833333333b2
0.3333333333ab + a2 + 0.25b2 = 0.5833333333b2

Factor a perfect square on the left side:
(a + 0.5b)(a + 0.5b) = 0.5833333333b2

Calculate the square root of the right side: 0.763762616b

Break this problem into two subproblems by setting 
(a + 0.5b) equal to 0.763762616b and -0.763762616b.

Subproblem 1

a + 0.5b = 0.763762616b Simplifying a + 0.5b = 0.763762616b Solving a + 0.5b = 0.763762616b 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 = 0.763762616b + -0.5b Combine like terms: 0.5b + -0.5b = 0.0 a + 0.0 = 0.763762616b + -0.5b a = 0.763762616b + -0.5b Combine like terms: 0.763762616b + -0.5b = 0.263762616b a = 0.263762616b Simplifying a = 0.263762616b

Subproblem 2

a + 0.5b = -0.763762616b Simplifying a + 0.5b = -0.763762616b Solving a + 0.5b = -0.763762616b 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 = -0.763762616b + -0.5b Combine like terms: 0.5b + -0.5b = 0.0 a + 0.0 = -0.763762616b + -0.5b a = -0.763762616b + -0.5b Combine like terms: -0.763762616b + -0.5b = -1.263762616b a = -1.263762616b Simplifying a = -1.263762616b

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.263762616b, -1.263762616b}

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