4a^2+ab-15b^2=0

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Solution for 4a^2+ab-15b^2=0 equation: 


Simplifying
4a2 + ab + -15b2 = 0

Reorder the terms:
ab + 4a2 + -15b2 = 0

Solving
ab + 4a2 + -15b2 = 0

Solving for variable 'a'.

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

Divide each side by '4'.
0.25ab + a2 + -3.75b2 = 0

Move the constant term to the right:

Add '3.75b2' to each side of the equation.
0.25ab + a2 + -3.75b2 + 3.75b2 = 0 + 3.75b2

Combine like terms: -3.75b2 + 3.75b2 = 0.00
0.25ab + a2 + 0.00 = 0 + 3.75b2
0.25ab + a2 = 0 + 3.75b2
Remove the zero:
0.25ab + a2 = 3.75b2

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.
0.25ab + a2 + 0.25b2 = 3.75b2 + 0.25b2

Combine like terms: 3.75b2 + 0.25b2 = 4b2
0.25ab + a2 + 0.25b2 = 4b2

Factor a perfect square on the left side:
(a + 0.5b)(a + 0.5b) = 4b2

Calculate the square root of the right side: 2b

Break this problem into two subproblems by setting 
(a + 0.5b) equal to 2b and -2b.


Subproblem 1
a + 0.5b = 2b

Simplifying
a + 0.5b = 2b

Solving
a + 0.5b = 2b

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 = 2b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = 2b + -0.5b
a = 2b + -0.5b

Combine like terms: 2b + -0.5b = 1.5b
a = 1.5b

Simplifying
a = 1.5b


Subproblem 2
a + 0.5b = -2b

Simplifying
a + 0.5b = -2b

Solving
a + 0.5b = -2b

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 = -2b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = -2b + -0.5b
a = -2b + -0.5b

Combine like terms: -2b + -0.5b = -2.5b
a = -2.5b

Simplifying
a = -2.5b


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {1.5b, -2.5b}

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