To solve this equation, we can use the Pythagorean identity which states that sin^2(x) + cos^2(x) = 1.

Given equation: sin^2(x) - 3cos^2(x) + 2sin(x)cos(x) = 0

Now, we substitute sin^2(x) with 1 - cos^2(x) in the equation:

(1 - cos^2(x)) - 3cos^2(x) + 2sin(x)cos(x) = 0

Expanding and simplifying:

1 - cos^2(x) - 3cos^2(x) + 2sin(x)cos(x) = 0
1 - 4cos^2(x) + 2sin(x)cos(x) = 0
1 - 4cos^2(x) + sin(2x) = 0

Now, we have the equation in terms of cosine and sine of angle 2x. Let y = 2x, then the equation becomes:

1 - 4cos^2(y/2) + sin(y) = 0

This equation might not have a simple solution using algebraic methods, and it may require numerical methods or graphical methods to find the solution.