Let us suppose that we have an equation

ax³ + bx² + cx + d = 0 .............1

Let it has a repeated root. Then we calculate derivative and equate to zero for same value of x i.e.

3ax² + 2bx + c = 0............2

Now multiply equation1 by 3 and multiply equation1 by x and equate them, we get

       3ax³ + 3bx² + 3cx + 3d = 3ax³ + 2bx² + cx

=>  3ax³ + 3bx² + 3cx + 3d - 3ax³ - 2bx² - cx = 0

=>  3bx² + 3cx + 3d - 2bx² - cx = 0

=>  bx² + 2cx + 3d = 0 

Now again take discrimanat and equate it to zero.

It gives us the value that x must have if it is a common root.