Which two numbers add up to and multiply to you me with this? multiplied 1 adds to 10

Begin by following the herd . . . call the two numbers 'x' and 'y' .

You said that  xy=1 ,  so  y = 1/x

You said that  x + y = 10, so          x + 1/x = 10

Multiply each side by 'x':                  x² + 1 = 10x

Subtract 10x from each side:   x² - 10x + 1  =  0

Use the quadratic formula to solve this, and the two solutions
that you get for  'x'  are also the two numbers  'x'  and  'y' .
They are

     9.89898
and
     0.10102

   (rounded)