(4 m)/(m-2)-13/(m-6) = 1/3

Multiply both sides by m-6:

(4 (m-6) m)/(m-2)-13 = (m-6)/3

Expand out terms on both sides:

(4 m^2)/(m-2)-(24 m)/(m-2)-13 = m/3-2

Write the left hand side as a single fraction:

(4 m^2-37 m+26)/(m-2) = m/3-2

Multiply both sides by m-2:

4 m^2-37 m+26 = 1/3 (m-2) m-2 (m-2)

Expand out terms on the right hand side:

4 m^2-37 m+26 = m^2/3-(8 m)/3+4

Subtract (m^2/3-(8 m)/3+4) from both sides:

(11 m^2)/3-(103 m)/3+22 = 0

Solve the quadratic equation by completing the square:

Divide both sides by 11/3:

m^2-(103 m)/11+6 = 0

Subtract 6 from both sides:

m^2-(103 m)/11 = -6

Add 10609/484 to both sides:

m^2-(103 m)/11+10609/484 = 7705/484

Factor the left hand side:

(m-103/22)^2 = 7705/484

Take the square root of both sides:

|m-103/22| = sqrt(7705)/22

Eliminate the absolute value:

m-103/22 = -sqrt(7705)/22 or m-103/22 = sqrt(7705)/22

Add 103/22 to both sides:

m = 1/22 (103-sqrt(7705)) or m-103/22 = sqrt(7705)/22

Add 103/22 to both sides:

m = 1/22 (103-sqrt(7705)) or m = 1/22 (103+sqrt(7705))