If $a < 0$, the positive root of the equation $x^2-2a|x-a| -3a^2=0$ is

x^2 - 2a|x-a| - 3a^2 = 0. If x >= a, x^2 - 2ax + 2a^2 - 3a^2 = 0, x^2 - 2ax - a^2 = 0. Roots: x = (2a +/- sqrt(4a^2 + 4a^2))/2 = a +/- a*sqrt(2). Since a < 0, a(1+sqrt(2)) is negative and a(1-sqrt(2)) is positive. Checking x < a: x^2 + 2ax - 2a^2 - 3a^2 = 0, x^2 + 2ax - 5a^2 = 0. Roots: x = (-2a +/- sqrt(4a^2 + 20a^2))/2 = -a +/- a*sqrt(6). Both are negative. The positive root is a(1-sqrt(2)).