To solve the equation sin^2(x) - 3cos^2(x) - 2sin(x)cos(x) = 0, we can use trigonometric identities to simplify the equation.

First, we know that sin(2x) = 2sin(x)cos(x). Therefore, we can rewrite the equation as:

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

Now, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) in the equation:

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

Next, we can simplify the equation by replacing sin(2x) with 2sin(x)cos(x):

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

Now, the equation resembles the original equation. Rearranging the terms, we get:

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

Factor out a -1 from the right side of the equation:

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

Divide both sides by -1 to simplify:

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

Therefore, sin^2(x) - 3cos^2(x) - 2sin(x)cos(x) = 0 simplifies to 3cos^2(x) + 2sin(x)cos(x) = 1

This is the simplified form of the given equation.